Advances in Space Research Journal ~ Volume 43, Issue 10, Pages 1471-1594, 15 May 2009

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Advances in Space Research 43 (2009) 1471–1478 www.elsevier.com/locate/asr

The acceleration of Anomalous Cosmic Rays by stochastic acceleration in the heliosheath L.A. Fisk *, G. Gloeckler Department of Atmospheric, Oceanic, and Space Sciences, University of Michigan, 2455 Hayward St., Ann Arbor, MI 48109-2143, USA Received 5 November 2008; received in revised form 9 February 2009; accepted 19 February 2009

Abstract Stochastic acceleration in the heliosheath appears to be a likely mechanism by which Anomalous Cosmic Rays (ACRs) are accelerated. However, most stochastic acceleration mechanisms are not appropriate. The energy density in the ACRs and in the interstellar pickup ions out of which the ACRs are accelerated greatly exceeds the energy density in the turbulence in the heliosheath. Thus, a traditional stochastic acceleration mechanism in which particles are accelerated by damping the turbulence will not work. A stochastic acceleration mechanism has been developed in which the total energy of the pickup ions and the ACRs is conserved. Energy is redistributed from the core pickup ions into a suprathermal tail to create the ACRs. A model for the acceleration of the ACRs in the heliosheath, based on this stochastic acceleration mechanism, is presented. The model provides reasonable ﬁts to the spectra of suprathermal particles and ACRs observed by Voyager. Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Anomalous Cosmic Rays; Acceleration; Heliosheath

1. Introduction Both Voyager spacecraft have now crossed the termination shock of the solar wind, where the supersonic solar wind ﬂow goes subsonic to begin the process of merging with the local interstellar medium (Stone et al., 2005, 2008; Decker et al., 2005, 2008; Burlaga et al., 2005, 2008; Gurnett and Kurth, 2005, 2008). The Voyagers are now penetrating into the heliosheath. Voyager 1 crossed in December 2004 at 94 AU from the Sun. Voyager 2 crossed in August 2007 at 83.7 AU. The two spacecraft were at substantially diﬀerent heliographic latitudes; Voyager 1 crossed at +34.3° and Voyager 2 at 27.5°. The shock crossings produced a major surprise. There was no evidence at either crossing of the acceleration of Anomalous Cosmic Rays (ACRs). Interstellar neutral gas is swept into the solar system by the motion of the Sun relative to the local interstellar medium. Once near the Sun *

Corresponding author. E-mail address: laﬁ[email protected] (L.A. Fisk).

the neutrals are ionized and then picked up by the outward ﬂowing solar wind, acquiring energies 1 keV nucleon1. The pickup ions are then convected into the outer solar system, where they are accelerated to energies of tens of MeV nucleon1, and form the ACRs (Fisk et al., 1974). It was expected that the ACRs would be accelerated at the termination shock for the simple reason that shocks are known to accelerate particles, and the termination shock is expected to be a relatively strong shock that surrounds the solar system (Pesses et al. (1981), Jokipii, 1990 and reference therein; Zank, 1999 and references therein). It is possible of course that the ACRs are accelerated at other locations on the termination shock, besides where the two Voyagers crossed. McComas and Schwadron (2006) have proposed a model in which the ACRs are accelerated along the ﬂanks of the termination shock, remote from the Voyagers. The Voyagers crossed closer to the nose of the heliosphere, in the direction the solar system is moving relative to the local interstellar medium. Given the turbulent nature of the heliospheric magnetic ﬁeld, and the random directions that result, it is not obvious why the termination

0273-1177/$36.00 Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2009.02.010

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shock on the ﬂanks should be a preferred acceleration site compared to the nose. It is also necessary in this model that the accelerated ACRs move freely along the magnetic ﬁeld in the heliosheath, with limited cross-ﬁeld diﬀusion, to account for the Voyager observations (Kota and Jokipii, 2008). It will be important to build a model that is consistent with the propagation properties both of ACRs and of comparable-rigidity galactic cosmic rays, which are believed to propagate in the heliosheath primarily by cross-ﬁeld diﬀusion. The alternative method for accelerating the ACRs is stochastic acceleration in the heliosheath. However, most stochastic acceleration mechanisms are not appropriate. In traditional stochastic acceleration mechanisms, and there are many in the literature, particles are accelerated by diffusing in velocity space (diﬀusion in velocity space is described in many basic plasma physics books, e.g., Bellan, 2006). The source of energy is the energy in the turbulence with which the particles are interacting; traditional stochastic acceleration mechanisms are a damping mechanism. In the heliosheath, however, the energy density in the turbulence is very much less than the energy density in the pickup ions and in the ACRs that are created from the pickup ions. The energy density or pressure in the pickup ions is dominant in the supersonic solar wind upstream from the termination shock. Voyager 2, which has a working solar wind plasma detector, observed that the ﬂow energy of the solar wind is converted into energy in the pickup ions at the termination shock, not to solar wind thermal energy (Richardson et al., 2008). Thus, the pickup ions are by far the dominant internal energy in the heliosheath, and, as we shall discuss, the energy density in the ACRs must be comparable to that of the pickup ions to explain the observed ACRs. There is simply not enough energy in the turbulence in the heliosheath to account for the energy in the ACRs, and a traditional stochastic acceleration mechanism in which the ACRs are accelerated by damping the turbulence will not work. Fisk and Gloeckler (2006, 2007, 2008) have developed a stochastic acceleration theory that seems ideally suited to accelerate the ACRs in the heliosheath. The theory is based on the premise that there are circumstances, spatial homogeneity, where it is reasonable to assume that the total energy in the core pickup ions and in the suprathermal tails, which are stochastically accelerated from the core, is a constant. This stochastic acceleration mechanism is thus a redistribution mechanism; energy is redistributed from the core into the tail particles. Energy is not extracted from the turbulence. In the heliosheath, where the dominant energy is contained in the pickup ions, not in the turbulence, only a redistribution mechanism will work. In addition, the stochastic acceleration mechanism of Fisk and Gloeckler (2006, 2007, 2008) applies in compressional turbulence, where there are random compressions and expansions of the thermal plasma, which contains the mass. The subsonic heliosheath contains ample compressional turbulence (Burlaga et al., 2005, 2008).

In Fisk and Gloeckler (2008) the equation describing the time evolution of their stochastic acceleration mechanism is derived. The solutions to the equation yield an equilibrium spectrum for the suprathermal tail on the distribution function that is a power law in particle speed with spectral index of 5. This is a robust result, independent of the governing parameters such as the particle spatial diﬀusion coeﬃcient. This result accounts for the fact that this spectral shape is observed to occur in many diﬀerent circumstances in the solar wind, in the quiet solar wind and in disturbed conditions downstream from shocks (e.g., Gloeckler et al., 2008). Of particular relevance to the discussion here, this is the spectral shape of the suprathermal tails on the pickup ion distributions observed by Voyager 1, now deep in the heliosheath (Decker et al., 2006; Hill et al., 2006). Voyager 2, at the time of the writing of this paper, still appears to be relatively close to the termination shock, where the conditions and the suprathermal particle spectra remain time varying (Decker et al., 2008). However, in all conditions in the supersonic solar wind and in the heliosheath where it is reasonable to consider that spatial homogeneity prevails, conditions that should particularly prevail in the deep heliosheath, the spectral shape of the suprathermal tails are observed to be power laws with spectral index of 5, consistent with the results of Fisk and Gloeckler (2006, 2007, 2008). Fisk and Gloeckler (2008) applied their stochastic acceleration theory to the suprathermal tails observed in the supersonic solar wind, where adiabatic deceleration due to the expansion of the mean solar wind ﬂow is important. They found energy and mass-to-charge dependencies of the rollovers in the spectra that occur at higher energies that agree well with observations from the Advanced Composition Explorer. They also found that the competition between the stochastic acceleration and the adiabatic deceleration limits the energy that suprathermal tail particles can achieve in the supersonic solar wind to about a few MeV nucleon1, consistent with the observations of Voyager in the outer heliosphere, upstream from the termination shock. In the subsonic heliosheath, there should be no or limited adiabatic deceleration. There is thus no competition to the stochastic acceleration mechanism of Fisk and Gloeckler (2008). The suprathermal tails on the pickup ions observed by Voyager should attain ever higher energies as the particles are convected into the heliosheath. The accelerated particles will then diﬀuse back in to be seen as the ACRs by Voyager, and eventually by other spacecraft in the inner heliosphere. In this paper, we apply the stochastic acceleration mechanisms developed in Fisk and Gloeckler (2008) to the acceleration of the ACRs in the heliosheath. We consider a simple model for acceleration in the heliosheath and show that the resulting spectra of the ACRs are in good agreement with the spectra observed by Voyager 1. We use the observations of multiple charge states of the higher energy ACRs to constrain the acceleration time, and thus our results are consistent with the multiple charge-state obser-

L.A. Fisk, G. Gloeckler / Advances in Space Research 43 (2009) 1471–1478

vations (Mewaldt et al., 1996). Finally, we comment on whether this theory can account for the observed behavior of the ACRs during the solar cycle. 2. The model Suprathermal tails on the pickup ion distributions, extending up to energies of about a few MeV nucleon1, are present in the supersonic solar wind in the outer heliosphere (Decker et al., 2005; Gloeckler et al., 2008). The spectra, expressed as distribution functions, are consistent with power laws in particle speed with spectral index of 5. Equivalently, the spectral index is 1.5 when the spectra are expressed as diﬀerential intensity, which is in more common use for higher energy particles. These particles are accelerated in crossing the termination shock, but the maximum energy of the suprathermal particles does not increase much beyond its upstream value. In the model of Fisk et al. (2006) for the termination shock, the pressure of the suprathermal tails behaves according to the Rankine–Hugoniot relationships, as do the core pickup ions. Initially behind the termination shock, there are time variations in the intensity and the spectral index of the suprathermal tails. However, as Voyager 1 has penetrated further into the heliosheath, in part because the termination shock at Voyager 1 appears to be moving inward, the spectra of the suprathermal tails settles into a remarkably constant power law spectrum with spectral index of 1.5 when expressed as diﬀerential intensity (Decker et al., 2006; Hill et al., 2006). Our model is then that the suprathermal tails present in the heliosheath are accelerated to higher energies to form the ACRs. We propose below that there is a prime acceleration region near the heliopause, where the suprathermal tails attain the highest energies, and that this is the source region of the ACRs. The particles accelerated in the prime acceleration region diﬀuse back into the heliosheath and ultimately into the inner heliosphere to be seen as ACRs. We use a simple spherically symmetric model for the heliosheath. The solar wind ﬂow is subsonic, and there is no adiabatic deceleration competing with our stochastic acceleration. The pressure in the interstellar pickup ions is dominant, and in this subsonic medium this pressure must be constant. The suprathermal tails and ultimately the ACRs are accelerated out of the pickup ions, and so the total pressure or energy density in the pickup ions, the suprathermal tails and the ACRs must be constant. We take the spatial diﬀusion coeﬃcient for all energetic particles, both the suprathermal tails and the ACRs, to be of the standard form, particle speed times a power law in particle rigidity, which for singly-charged pickup ions and in terms of particle kinetic energy, becomes j ¼ jo Aa Eðaþ1Þ=2 :

ð1Þ

Here, jo is a constant, i.e. we neglect here for simplicity any spatial dependence in the diﬀusion coeﬃcient; A is the mass number, and E is particle kinetic energy, measured in units

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of MeV nucleon1. The power law index of the rigidity dependence of j is a, which is an adjustable parameter that will be chosen to ﬁt the observations. We also take the mean square speed of the compressional turbulence in the heliosheath, du2, to be constant. 2.1. The basic equation and the resulting spectra We use the equation for stochastic acceleration derived in Fisk and Gloeckler (2008). The steady state equation for the distribution function, f, for the stochastic acceleration in a spherically symmetric heliosheath, is then @ 5 @ du2 @ 5 v ðv f Þ : ð2Þ u ðv f Þ ¼ v @r @v 9j @v Here, u is the radial component of the solar wind speed; v is particle speed; r is heliocentric radial distance. Eq. (2) has a straightforward solution " # 9j 1 5 : ð3Þ f ¼ fo v exp 2 ð1 þ aÞ du2 strans Here, the quantity strans satisﬁes Z r dstrans 1 1 ¼ or strans ¼ dr; dr u ro u

ð4Þ

and represents the transit time at the solar wind speed from where the particles are ﬁrst able to be accelerated to where they obtain their highest energies. We can also express our resulting spectrum in terms of the diﬀerential intensity, j, which is more common in dealing with energetic particle spectra, or " # 9j 1 : ð5Þ j / E1:5 exp ð1 þ aÞ2 du2 strans We should note that there are various means by which strans could be limited. For example, particles could leak from the acceleration region by diﬀusion, in which case we can readily show that there is a maximum value for strans equal to strans;max ¼

3g ; ð1 þ aÞdu

ð6Þ

where g is the characteristic scale length for the diﬀusive escape. 2.2. The solar wind speed in the heliosheath Consider the solar wind speed in the heliosheath. In a simple spherically symmetric heliosheath, the solar wind velocity ﬂows radially and falls oﬀ as 1/r2, yielding a constant solar wind density and no adiabatic deceleration. However, if the heliopause is not inﬁnitely distant, then the solar wind ﬂow needs to turn and ﬂow parallel to the heliopause. The ﬂow can still remain relatively incompressible ($ u = 0), with no adiabatic deceleration. However, the radial component of the ﬂow goes to zero at the

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heliopause. If the accelerated particles ﬂow freely along the primarily azimuthal magnetic ﬁeld, or are capable of rapid spatial diﬀusion, the gradients of the accelerated particles should be small in the azimuthal direction, and Eq. (2), which is for spherically symmetric conditions, should still apply. The solar wind speed in Eq. (2) has then to be interpreted only as the radial component of the solar wind ﬂow, and to go to zero at the heliopause. For simplicity we take the radial solar wind speed in the heliosheath to be 2 rts k : ð7Þ ur ¼ uts 2 exp r ðrhp rÞ Here, ts denotes termination shock and hp, the heliopause; k is the characteristic distance at which the radial solar wind speed begins to decline to zero at the heliopause. Consider, for example, that rts = 90 AU (the termination shock seen by Voyager 1 has moved inward); uts = 135 km s1, as seen on Voyager 2 (Richardson et al. 2008); rhp = 140 AU, and k = 8 AU. Then the transit time from the termination shock in years, given in Eq. (4), plotted versus the radial distance in AU, is shown in Fig. 1. Clearly, the particles spend most of their time near the heliopause. This is where the prime acceleration occurs, and where the highest energy particles are created; where the ACRs are created.

mum energy spectrum near the heliopause can be relative to the local suprathermal tail. Fisk and Gloeckler (2007) showed that the pressure in the tail relative to the pressure in the core particles should be given by P tail 2 b ¼ ; P core 5 ð1 þ bÞ

where b is the maximum relative spatial variation in the core or in the tail pressure. They found observational support for this expression; in the slow solar wind in the inner heliosphere, b 0.66, and Ptail/Pcore 0.16. Fisk et al. (2006) found that Ptail/Pcore 0.32 immediately upstream of the termination shock, as observed by Voyager 1. However, Gloeckler and Fisk (2006) then showed that the suprathermal particles upstream of the termination shock are due to beams of downstream particles, and the background spectrum which is actually accelerated at the shock is a factor of 3 lower than the average of the beam spectra. That being the case, a value of Ptail/Pcore 0.16 is probably a reasonable result upstream of the termination shock, and since the core and the tail pressures are each increased the same, according to the Rankine–Hugoniot relationship, this ratio should hold downstream as well. We take then the total pressure in the pickup ions immediately downstream from the termination shock to be P tot;ts ¼ 7:3P tail;ts ;

2.3. The normalization of the maximum energy spectrum

ð8Þ

ð9Þ

Consider the normalization factor for the spectrum in Eq. (5), evaluated at the maximum value of strans, i.e. the spectrum that attains the highest energy particles, the ACRs. This spectrum should occur near the heliopause. The basic requirement in the heliosheath is that the pressure in the pickup ions remains constant. The pickup ions are the dominant pressure and in the subsonic heliosheath the pressure must be constant. This requirement places a constraint on how much larger the intensity of the maxi-

where the subscript ts refers to near the termination shock. The criterion in Eq. (8) was derived by assuming that there was a ﬁxed threshold between the core and the tail, and that only those core particles that were adiabatically compressed and obtain energies above the threshold could ﬂow into the tail (Fisk and Gloeckler, 2007). The location of this threshold was established in Fisk and Gloeckler (2008) (see Eq. (19)) to be j r du ; ð10Þ l2 3

Fig. 1. Transit time in years versus the distance from the termination shock.

where l is the characteristic scale length of the compressional turbulence; e.g., the correlation length of the compressional turbulence. Note that |$ du| du/l. In principle, then, the threshold energy can decrease with decreasing l, provided that j does not scale as l. In our model for the heliosheath, the radial solar wind speed decreases as the ﬂow approaches the heliopause. The scale length l is in the direction of du, which compresses the magnetic ﬁeld and thus is in a direction normal to the magnetic ﬁeld and in the radial direction. The scale length l should thus decrease as the radial solar wind speed decreases near the heliopause, and the threshold in the source region of the ACRs, given by Eq. (10), will be lower. That is, a larger fraction of the pressure in the core can be placed in the tail than is predicted by Eq. (8). In Eq. (8), compressions raise the energy of the particles to exceed the threshold, and only a certain fraction of the energy can be raised above the threshold. However, if the thresh-

L.A. Fisk, G. Gloeckler / Advances in Space Research 43 (2009) 1471–1478

old is lowered, there is no limit to how much core pressure can be deposited in the tail. In principle the core could vanish. In the source region of the ACRs there is only the core and the ACRs; the suprathermal tail is the ACRs. We take n to be the ratio of the tail to the core pressure, and thus the total pressure in the pickup ions in the source region s is P total;s ¼ ð1 þ nÞP ACR;s :

ð11Þ

The pressure of the ACRs near the termination shock also enters into the pressure balance required in the heliosheath. However, the pressure in the ACRs near the termination shock can be neglected since these particles are modulated, which also reduces the roll-over energy and the intensity. Our requirement of constant pressure in the pickup ions throughout the heliosheath thus becomes 7:3P tail;ts ¼ ð1 þ nÞP ACR;s :

ð12Þ

Some of the pressure increase in PACR in the source region, compared to Ptail,ts, results because the ACRs in the source region obtain higher energies than does the tail nearer to the termination shock. With a spectral index of 1.5 (or 5 when expressed as a distribution function) the pressure increases as the log of the energy. We also concluded that the low-energy threshold in the source region is lower than near the termination shock; e.g., let us take it to be a factor of 2 lower. Typical values that we will use below to ﬁt the spectra observed by Voyager 1 are Eo = 160; Eo,tail,ts = 17; Eth = 0.008 (recall, E is in units of MeV nucleon1). Thus, the pressure increase in the ACRs in the source region due solely to the increase in energy is ln½2Eo =Eth ¼ 1:4: ln½Eo;tail;ts =Eth

ð13Þ

The remaining increase in pressure in the ACRs in the source region, required to satisfy Eq. (12), is due to an increase in the normalization factor on Eq. (5). There is also a possibility that the normalization factor is diﬀerent for diﬀerent species. Since the diﬀusion coeﬃcient in Eq. (10) is a function of rigidity, the threshold energy can be lower for higher mass-to-charge particles, suggesting that the normalization factor will be larger for these particles as well. The above argument is best applied in reverse: we use our model to ﬁt the observations of Voyager 1 and from these ﬁts we infer the normalization factors in the source region for the diﬀerent species of ACRs. Gloeckler et al. (2009) performed this exercise using a model similar to the one developed here and found that the increase in the normalization factor for ACRs in the source region, as compared to the normalization factor on the suprathermal tails seen at the location of Voyager 1, is a factor 2.5, and not strongly dependent on ACR species. This result is similar to the ﬁts to the ACR spectra presented below. Thus, from Eqs. (12) and (13), we infer that the pressure in the ACRs in the source region is comparable to the pressure in the core pickup ions. This result

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strongly supports our conclusion that there is insuﬃcient energy in the turbulence in the heliosheath to accelerate the ACRs, and thus a stochastic acceleration mechanism in which particles are accelerated by damping turbulence is inappropriate. 2.4. Conditions in the acceleration region near the heliopause and constraints from observations of multiple charge states We expect that the normalization factor on the ACR spectrum in the source region near the heliopause is a factor of 2.5 larger than the normalization factor on the spectrum of the local suprathermal tail. We also expect that the spectrum near the heliopause obtains the highest possible energies since this region, with its slow radial solar wind speeds, can contain the longest transit times. We need, however, to consider only the transit time near the heliopause since this is the region where the threshold is lowered and where there is the extra injection of particles by a factor 2, above the lower energy threshold, i.e., most of the particles accelerated near the heliopause were injected in this region.There is a constraint on the maximum allowable acceleration time, or in our case the maximum allowable transit time for particles injected near the heliopause. The presence of multiple charge states of O at energies above 20 MeV nucleon1 as seen by SAMPEX, and their absence below these energies, allowed Mewaldt et al. (1996) to conclude that the acceleration time of 10 MeV nucleon1 O had to be 1 year. All particles being accelerated by our mechanism experience the same transit time. We thus have a constraint that the transit time in the prime acceleration region, from where the threshold on the core is lowered and a factor of 2 times more particles are injected, to where the maximum energies are achieved, is 1 year. From Fig. 1, the prime acceleration region could be within a few AU of the heliopause. In either case the limit on the transit time could be determined by Eq. (6). With this constraint on the transit time we also have a constraint on the diﬀusion coeﬃcient and the turbulent speeds in the prime acceleration region near the heliopause. We assume that this region near the heliopause may be different than the rest of the heliosheath, e.g., since there are strong shear ﬂows at the heliopause, the region could experience Kelvin-Helmholtz instabilities and be more turbulent. We are free then to choose the diﬀusion coeﬃcient and the turbulent speeds in the prime acceleration region, separate from the diﬀusion coeﬃcient and turbulent speeds that prevail elsewhere in the heliosheath. We could also adjust the rigidity dependence of the diﬀusion coeﬃcient, i.e., the value of a in Eq. (1), to be diﬀerent near the heliopause, but we will not introduce this additional complication. We have then a single source region for the ACRs. It is near the heliopause where the radial solar wind speed is very low, the threshold between the core and the tail has been lowered, and diﬀusion coeﬃcient and turbulent speeds are conducive to acceleration.

L.A. Fisk, G. Gloeckler / Advances in Space Research 43 (2009) 1471–1478

j ¼ jo E1:5

uj ¼ j

dj : dr

ð14Þ

Or with j given in Eq. (1), u given in Eq. (7), and the source spectrum given in Eq. (5), we ﬁnd that the intensity spectrum of the ACRs at r is given by j ¼ jo E1:5

" ðaþ1Þ=2 # uts rv a E exp A exp : Eo jo Aa Eð1þaÞ=2

ð15Þ

Here, v¼

1 r

Z

rhp r

2 rts k dr0 ; exp ðrhp r0 Þ r 02

ð16Þ

and 2

Eoðaþ1Þ=2 ¼

ð1 þ aÞ du2hp strans;acc : 9jo;hp

ð17Þ

We have allowed for the fact that the conditions near the heliopause may be diﬀerent from those elsewhere in the heliosheath, and labeled the parameters describing conditions near the heliopause by the subscript hp. We note also that the diﬀerent choice for the diﬀusion coeﬃcient near the heliopause does not aﬀect the modulation since the radial solar wind speed is eﬀectively zero there. The value of jo can be determined by ﬁtting the observations and in turn will set the threshold between the core and the tail in the source region of the ACRs. Note that in determining the radial dependence of j in Eq. (16), the radial dependence of v needs to be taken into account. 2.6. Fitting the observations We ﬁrst use the gradient of He observed by Voyager in the heliosheath to specify the spatial diﬀusion coeﬃcient. At 16 MeV nucleon1, Stone et al. (2008) ﬁnds the gradient of He to be 5% per AU. We assume that this gradient exists at a reference location, rref, equal to 100 AU; we take the solar wind speed at the reference location to be uref. We can then use Eq. (14) to specify jo as jo ¼

uref rref 54

ð2aþ1Þ

:

ð18Þ

Eq. (18) can then be substituted into Eq. (15) to yield the ﬁnal result for j for ACRs as

# " ð1þaÞ=2 # 5 4ð2aþ1Þ rv uts a E exp a ð1þaÞ=2 exp A : rref uref Eo AE ð19Þ

We will consider the Voyager 1 observations at r = 100 AU, or equivalently at rref. The solar wind speed, by Eq. (7), should decrease between rts and rref, or if rts = 90 AU for Voyager 1, and r = 100 AU, then, uts/uref = 1.23. If we then use the same parameters as in Fig. 2, we ﬁnd that v = 0.14. Finally, Eq. (19) becomes at 100 AU, j ¼ jo E1:5

"

# " ð1þaÞ=2 # 0:85 4ð2aþ1Þ a E exp a ð1þaÞ=2 exp A : Eo AE

ð20Þ

In Fig. 2, we compare the solutions predicted by Eq. (20) with the ACR spectra observed by the LECP instrument on Voyager 1 (Decker et al., 2005, 2008; Gloeckler et al., 2008, 2009). The parameters are Eo = 160 and a = 0.87; the normalization constants are chosen to ﬁt the observations, and all elements are a factor of 2 above the local suprathermal tails at 100 AU. Also shown in Fig. 2 are the local suprathermal tail spectra predicted by Eq. (3). Here the parameters are Eo = 17 and the same a = 0.87. The combined spectra yield a reasonable ﬁt to the Voyager observations. Note that the spectra of 6 diﬀerent elements are well ﬁt with, in the case of the ACRs, only two parameters, Eo and a. If we use uts = 135 km s1, the value observed at the Voyager 2 crossing (Richardson et al., 2008), and the radial dependence of the radial solar wind speed in Eq. (7), we ﬁnd from Eq. (18) that

Heliosheath 100

H

Voyager 1 LECP 2005.0 - 2008.0

He 10-2

10-4

10-6

N

O

Ne

Fig 2 dj/dE Sum 2005.0-2007.200 ACR paper.qpc

Since the ACRs come from a single source region, the modulation problem becomes very simple. The modulation in the heliosheath is simple convection-diﬀusion modulation, without adiabatic deceleration. The governing equation is

"

-1

2.5. Modulation of the ACRs

Differential Intensity (s cm 2 sr MeV/nuc)

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Ar

10-8

10-10 10-1

100

101

102

Energy/nucleon (MeV/nuc) Fig. 2. Model ﬁts to the spectra of H, He, O, N, Ne, and Ar observed by the LECP instrument on Voyager 1 at 100 AU in the heliosheath. Data are from Decker et al. (2005, 2008) and Gloeckler et al. (2009).

L.A. Fisk, G. Gloeckler / Advances in Space Research 43 (2009) 1471–1478

jo ¼ 5:3 1019 cm2 s1 :

ð21Þ

Also, we assume that the prime acceleration region near the heliopause is more turbulent than elsewhere in the heliosheath and take du 100 km s1, as opposed to 35 km s1 observed by Voyager 2 (Richardson et al., 2008). Then, with the constraint that the transit time in the prime acceleration region should be 1 year, and with Eo = 160 (in MeV nucleon1) and a = 0.87, we ﬁnd from Eq. (17) that jo;hp ¼ 1019 cm2 s1 :

ð22Þ

Thus, the diﬀusion coeﬃcient in the prime acceleration region near the heliopause is a factor of 5 times smaller than elsewhere in the heliosheath. Again, we are free to choose a smaller value for the diﬀusion coeﬃcient in the acceleration region, without regard to the impact on the modulation, since the radial solar wind speed is eﬀectively zero there. Finally, from Eq. (6), with du 100 km s1 and the transit time of 1 year, we ﬁnd that g = 12 AU, i.e., the principal escape by diﬀusion is across the heliopause. It should be emphasized that all of these values for the parameters describing the conditions near the heliopause depend on the constraint that the acceleration time of the ACRs is limited to 1 year. The general concepts of the theory for the acceleration of the ACRs would not change if there is a diﬀerent limitation of the acceleration time; however, the ﬁnal values for the governing parameters would be diﬀerent. 2.7. Dependence on the solar magnetic cycle There is another constraint on acceleration of ACRs that is worth commenting upon: the dependence of the ACRs on the solar magnetic cycle. Full consideration of this constraint will require more sophisticated models than the simple analytic formulae that we have developed. We will leave that to later work. ACRs are known to vary with the solar magnetic cycle, with the largest intensities occurring in the so-called A < 0 portion of the solar magnetic cycle, when the heliospheric magnetic ﬁeld is inward toward the Sun in the northern hemisphere. Jokipii and co-workers (Jokipii, 1982, 1986; Jokipii and Giacalone, 1998) have argued that this dependence is the result of gradient drifts along the termination shock. During the A > 0 portion of the solar magnetic cycle, the ACRs drift toward the pole along the termination shock, then inward into the heliosphere, and downward in latitudes, and then outward again along the current sheet. The maximum of the intensity of the ACRs is then near the poles. For the A < 0 cycle, when the heliospheric ﬁeld is inward in the northern hemisphere, the directions are opposite, and the peak in the ACR intensity should be at low latitudes, which is seen as higher ACR intensities by Voyager and at Earth. Particles drift along the termination shock because of the sharp gradient in the magnetic ﬁeld. The heliospheric

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magnetic ﬁeld declines in the supersonic solar wind, as 1/r. The ﬁeld strength always increases when the shock is crossed. Hence the gradients are in opposite direction, and the resulting drifts in the supersonic solar wind and along the termination shock are always opposite to each other. In our model for the heliosheath, the radial velocity of the solar wind slows down as the ﬂow approaches the heliopause, and the ﬂow turns to ﬂow parallel to the heliopause. The ﬂow could remain incompressible, and this is the normal assumption since the gas is subsonic. However, some compression is allowed. The actual requirement is that the pressure is constant, but the pressure is all in the pickup ions. Thus, it is possible that the density of the solar wind, and thus the frozen-in magnetic ﬁeld increases near the heliopause. We have then exactly the same gradient in the ﬁeld strength as occurs near the termination shock. The drifts are slower since the ﬁeld increase occurs over a longer distance than the abrupt increase that occurs at the termination shock. Nonetheless, the drifts will carry the ACRs towards the pole in the A > 0 cycle and visa versa in the A < 0 cycle. 3. Concluding remarks We have developed a theory for the stochastic acceleration of ACRs in the heliosheath that is consistent with the required source of energy, viz. the pickup ions themselves, and that provides a reasonable ﬁt to the spectra of ACRs observed by Voyager 1. We have limited our ﬁts to Voyager 1 data, since Voyager 1, unlike Voyager 2, is now deep into the heliosheath, and a steady-state model, such as the one we have constructed, should be valid. Clearly, as Voyager 2 penetrates further into the heliosheath, it will provide further tests of our model. There are of course limitations to our analytic model, which can only be addressed with a full numerical simulation. One of these limitations is the assumption of spherical symmetry. We assumed that the accelerated pickup ions have only small azimuthal gradients in the heliosheath, and thus only the radial solar wind ﬂow is important in determining the transit time and thus the acceleration time of the ACRs. This assumption, as well as our assumption that the acceleration of the ACRs occurs primarily near the heliopause, should be valid in the direction of motion of the solar system relative to the local interstellar medium, the so-called nose region, which is generally in the region where the Voyagers are located. Here, the heliosheath should be its narrowest, and thus radial gradients are more important than azimuthal gradients. Further, there is more likely to be a preferred acceleration region near the heliopause, where the drop in the solar wind speed occurs in a relatively narrow band, creating a region of preferred injection and acceleration. On the ﬂanks of the heliosheath, however, conditions are likely to be diﬀerent; in particular, the heliosheath will be wider, resulting in a more gradual decline in the radial solar wind ﬂow, and perhaps diﬀerent

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conditions near the heliopause. In many ways, we might expect that the heliosheath in the nose region is a better location for accelerating ACRs to high energies than on the ﬂanks. This will of course introduce azimuthal variations in the production of ACRs. Such variations are unlikely to be observable in the inner heliosphere, since the Parker spiral of the heliospheric magnetic ﬁeld in the supersonic solar wind circles the Sun many times between the inner heliosphere and the termination shock, and should smooth out any variations in the production of ACRs in the heliosheath. Acknowledgement This work was supported in part by the Voyager Guest Investigation grant NNX07AH, and by ACE data analysis contract, 44A-1080828. References Bellan, P.M. Fundamentals of Plasma Physics. Cambridge University Press, Cambridge, 2006. Burlaga, L.F., Ness, N.F., Acun˜a, M.H., Lepping, R.P., Connerney, J.E.P., Richardson, J.D. Magnetic ﬁelds at the solar wind termination shock. Nature 454, 75–77, 2008. Burlaga, L.F., Ness, N.F., Acun˜a, M.H., Lepping, R.P., Connerney, J.E.P., Stone, E.C., McDonald, F.B. Crossing the termination shock in the heliosheath: magnetic ﬁelds. Science 309, 2027–2029, 2005. Decker, R.B., Krimigis, S.M., Roelof, E.C., Hill, M.E., Armstrong, T.P., Gloeckler, G., Hamilton, D.E., Lanzerotti, L.J. Voyager 1 in the foreshock, termination shock and heliosheath. Science 309, 2020–2024, 2005. Decker, R.B., Krimigis, S.M., Roelof, E.C., Hill, M.E., Armstrong, T.P., Gloeckler, G., Hamilton, D.E., Lanzerotti, L.J. Mediation of the solar wind termination shock by non-thermal ions. Nature 454, 67–70, 2008. Decker, R.B., Roelof, E.C., Krimigis, S.M., Hill, M.E. Low-energy ions near the termination shock, in: J. Heerikhuisen et al. (Eds.), AIP Conf. Proc. 858, Physics of the Inner Heliosheath, Danvers, MA, AIPC, p. 73–78, 2006. Fisk, L.A., Gloeckler, G. The common spectrum for accelerated ions in the quiet-time solar wind. Astrophys. J. 640, L79–L82, 2006. Fisk, L.A., Gloeckler, G. Thermodynamic constraints on stochastic acceleration in compressional turbulence. Proc. Natl. Acad. Sci. 104, 5749–5754, 2007. Fisk, L.A., Gloeckler, G. Acceleration of suprathermal tails in the solar wind. Astrophys. J. 686, 1466–1473, 2008. Fisk, L.A., Gloeckler, G., Zurbuchen, T.H. Acceleration of low-energy ions at the termination shock of the solar wind. Astrophys. J. 644, 631– 637, 2006. Fisk, L.A., Kozlovsky, B., Ramaty, R. An interpretation of the observed oxygen and nitrogen enhancements in low energy cosmic rays. Astrophys. J. Lett. 190, L35–L37, 1974.

Gloeckler, G., Fisk, L.A. Anisotropic beams upstream of the termination shock of the solar wind. Astrophys. J. 648, L63–L66, 2006. Gloeckler, G., Fisk, L.A., Mason, G.M., Hill, M.E. Formation of power law tail with spectral index 5 inside and beyond the heliosphere, in: Li, G., Hu, Q., Verhoglyadora, O., Zank, G.P., Lin, R.P., Luhmann, J. (Eds.), AIP Conf. Proc. 1039, Particle Acceleration and Transport in the Heliosheath and Beyond, 367–374, 2008. Gloeckler, G., Fisk, L.A., Geiss, J., Hill, M.E., Hamilton, D.C., Decker, R.B., Krimigis, S.M. Composition of interstellar neutrals and the origin of anomalous cosmic rays. Space Sci. Rev. 143, 163– 175, 2009. Gurnett, D.A., Kurth, W.S. Electron plasma oscillations upstream of the solar wind termination shock. Science 309, 2025–2027, 2005. Gurnett, D.A., Kurth, W.S. Intense plasma waves at and near the solar wind termination shock. Nature 454, 78–80, 2008. Hill, M.E., Decker, R.B., Roelof, E.C., Krimigis, S.M., Gloeckler, G. Heliosheath particles anomalous cosmic rays and a possible ‘‘third source” of energetic ions, in: Heerikhuisen, J. et al. (Eds.), AIP Conf. Proc. 858 Physics of the Inner Heliosheath. AIPC, Danvers, MA, pp. 98–103, 2006. Jokipii, J.R. Particle drift, diﬀusion and acceleration at shocks. Astrophys. J. 255, 716–720, 1982. Jokipii, J.R. Particle acceleration at a termination shock. I – application to the solar wind and the anomalous component. J. Geophys. Res. 91, 2929–2932, 1986. Jokipii, J.R. The anomalous component of cosmic rays, in: Grzedielski, S., Page, D.E. (Eds.), COSPAR Colloq., Physics of the Outer Heliosphere. Pergamon, Elmsford, pp. 169–178, 1990. Jokipii, J.R., Giacalone, J. The theory of anomalous cosmic rays. Space Sci. Rev. 83, 123–136, 1998. Kota, J., Jokipii, J.R. 2008. Anomalous cosmic rays in the heliosheath: simulation with a blunt termination shock, in: Li, G., Hu, Q., Verhoglyadora, O., Zank, G.P., Lin, R.P. Luhmann, J. (Eds.), AIP Conf. Proc. 1039, Particle Acceleration and Transport in the Heliosheath and Beyond, 397–403, 2008. McComas, D.J., Schwadron, N.A. An explanation for the Voyager paradox: particle acceleration at a blunt termination shock. Geophys. Res. Lett. 33, L04102, doi:10.1029/2005GL025437, 2006. Mewaldt, R.A., Selesnick, R.S., Cummings, A.C., Stone, E.C. Evidence for multiply charged anomalous cosmic rays. Astrophys. J. Lett., L43– L46, 1996. Pesses, M.E., Jokipii, J.R., Eicher, D. Cosmic ray drift, shock wave acceleration, and the anomalous component of cosmic rays. Astrophys. J. Lett. 246, L85–L88, 1981. Richardson, J.D., Kasper, J.C., Wang, C., Belcher, W., Lazarsu, A.J. Cool heliosheath plasma and deceleration of the upstream solar wind at the termination shock. Nature 454, 63–66, 2008. Stone, E.C., Cummings, A.C., McDonald, F.B., Heikkila, B.C., Lal, N., Webber, W.R. Voyager 1 explores the termination shock region and heliosheath beyond. Science 309, 2017–2020, 2005. Stone, E.C., Cummings, A.C., McDonald, F.B., Heikkila, B.C., Lal, N., Webber, W.R. An asymmetric solar wind termination shock. Nature 454, 71–74, 2008. Zank, G. Interaction of the solar wind with the local interstellar medium: a theoretical perspective. Space Sci. Rev. 89, 413–688, 1999.

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Advances in Space Research 43 (2009) 1479–1483 www.elsevier.com/locate/asr

Coronal fast wave trains of the decimetric type IV radio event observed during the decay phase of the June 6, 2000 ﬂare H. Me´sza´rosova´ a,*, H.S. Sawant b, J.R. Cecatto b, J. Ryba´k c, M. Karlicky´ a, F.C.R. Fernandes d, M.C. de Andrade b, K. Jirˇicˇka a a Astronomical Institute, Czech Academy of Sciences, CZ-25165 Ondrˇejov, Czech Republic National Space Research Institute (INPE), Ave. dos Astronautas 1758, 1221-0000 Sa˜o Jose´ dos Campos, SP, Brazil c Astronomical Institute, Slovak Academy of Sciences, SK-05960 Tatranska´ Lomnica, Slovak Republic d Institute of Research and Development (IP&D – UNIVAP), Ave. Shishima Hifum 2911, Urbanova, 12244-000 Sa˜o Jose´ dos Campos, SP, Brazil b

Received 27 October 2008; received in revised form 23 January 2009; accepted 27 January 2009

Abstract The 22 min long decimetric type IV radio event observed during the decay phase of the June 6, 2000 ﬂare simultaneously by the Brazilian Solar Spectroscope (BSS) and the Ondrˇejov radiospectrograph in frequency range 1200–4500 MHz has been analyzed. We have found that the characteristic periods of about 60 s belong to the long-period spectral component of the fast wave trains with a tadpole pattern in their wavelet power spectra. We have detected these trains in the whole frequency range 1200–4500 MHz. The behavior of individual wave trains at lower frequencies is diﬀerent from that at higher frequencies. These individual wave trains have some common as well as diﬀerent properties. In this paper, we focus on two examples of wave trains in a loop segment and the main statistical parameters in their wavelet power and global spectra are studied and discussed. Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Sun; Corona; Flares; Radio radiation; MHD waves

1. Introduction It has been theoretically predicted (Roberts et al., 1983, 1984) that periodicity of fast magnetoacoustic modes can be modiﬁed by the time evolution of an impulsively generated signal. An obvious source of such an impulsive disturbance is a ﬂare (providing either a single or multiple source of disturbances). These fast magnetoacoustic waves are trapped in regions (e.g. coronal loop) with a high density (i.e. with a low Alfve´n speed). These regions are acting as waveguides. The impulsively generated (propagating) wave

*

Corresponding author. Tel.: +420 323 620155; fax: +420 323 620210. E-mail addresses: [email protected] (H. Me´sza´rosova´), sawant@das. inpe.br (H.S. Sawant), [email protected] (J.R. Cecatto), [email protected] (J. Ryba´k), [email protected] (M. Karlicky´), [email protected] (F.C.R. Fernandes), [email protected] (M.C. de Andrade), [email protected] (K. Jirˇicˇka).

in a density coronal waveguide exhibits several phases: (1) periodic phase (long-period spectral components arrive as the ﬁrst at the observation point), (2) quasi-periodic phase (as both long and short-period spectral components arrive and interact), and ﬁnally (3) decay phase (as the signal passes). The quasi-periodic phase is generally much stronger in amplitude and shorter in ‘periodicity’ than the earlier periodic phase. The numerical simulation of characteristic time (wavelet) signatures of impulsively generated fast magnetoacoustic wave trains propagating along a coronal loop with diﬀerent ratios of the density contrast has been studied by Nakariakov et al. (2004). It was found that the dispersive evolution of fast wave trains leads to the appearance of characteristic tadpole wavelet signature where a narrow-spectrum tail precedes a broadband head. Such tadpole signatures were observed in solar eclipse data (Katsiyannis et al., 2003; Williams et al., 2001). Now, for

0273-1177/$36.00 Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2009.01.032

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the ﬁrst time, these tadpole wavelet signatures of impulsively generated fast magnetoacoustic wave trains observed in decimetric type IV radio event are presented. 2. Observation and results The June 6, 2000 ﬂare (classiﬁed as X2.3, GOES X-ray maximum at 15:25 UT) was observed during 14:58– 17:00 UT in the active region NOAA AR 9026. In Ha the ﬂare has the importance 2B. We have observed a 22 min long (15:40:05–16:02:00 UT) decimetric type IV radio event during the decay phase of this ﬂare recorded simultaneously by the Brazilian Solar Spectroscope (BSS, frequency range 1200–1700 MHz, time and frequency resolution is 50 ms and 5 MHz, respectively) and by the Ondrˇejov radiospectrograph (frequency range 2000– 4500 MHz, time and frequency resolution is 100 ms and 10 MHz, respectively). The radio spectra for the whole time interval are shown in Fig. 1. The high time and frequency resolutions of both instruments enable us to investigate the spectra in detail. Time series of these spectra have been analyzed in their power and global wavelet spectra at individual frequencies in the range 1200–4500 MHz.

For the analysis, we have divided the interval under study (15:40:05–16:02:00 UT) into four subintervals A–D (Table 1) and we have recognized tadpole wavelet signatures in all these subintervals. An example of the characteristic tadpole pattern (time subinterval C) is shown in Fig. 2. The top panel shows the time series at the frequency 1395 MHz and the middle panel exhibits the corresponding wavelet power spectrum with tadpole pattern as the signature of a coronal fast wave train. The lighter area indicates greater power in the wavelet power spectrum and the hatched region belongs to the cone of inﬂuence (COI) where edge eﬀects become important due to dealing with ﬁnite-length time series. The solid contour shows the 95% conﬁdence level. In the analysis of each of time series only these contoured regions which exist outside the COI have been considered as valid. The long-period spectral component (tadpole tail) has characteristic period P = 63.8 s. The range of periods at the point of the maximal extension of the tadpole head is 39.5–70.1 s. The bottom panel shows the global wavelet spectrum with the characteristic period P = 63.8 s above the 95% global signiﬁcance level (horizontal line). Some examples of tadpoles at diﬀerent frequencies are presented in Figs. 3 and 4.

Fig. 1. Left panels: 22 min long decimetric type IV radio spectrum (the highest intensity in black) observed during the decay phase of the June 6, 2000 ﬂare recorded simultaneously by the BSS (1200–1700 MHz) and by the Ondrˇejov radiospectrograph (2000–4500 MHz). Right panels: characteristic ﬂux time series at six selected frequencies.

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Table 1 Time subintervals of broadband pulsations observed during the decay phase of the June 6, 2000 ﬂare in the frequency range 1200–4500 MHz. Time subinterval

Start (UT)

End (UT)

A B C D

15:40:05 15:45:34 15:51:03 15:56:32

15:45:33 15:51:02 15:56:31 16:02:00

Fig. 2. Example of a tadpole wavelet pattern of a coronal fast wave train. Top panel: time series at the frequency 1395 MHz. Middle panel: corresponding wavelet power spectrum with a tadpole signature. The lighter area indicates a greater power in the wavelet power spectrum and the hatched region belongs to the cone of inﬂuence (COI). The solid contour shows the 95% conﬁdence level. The range of powers (grey scale) is presented on right side of the spectrum. The long-period spectral component (tadpole tail) has characteristic period P = 63.8 s. Bottom panel: global wavelet spectrum with the characteristic period P = 63.8 s above the 95% global signiﬁcance level (horizontal line).

Series of two tadpoles (time subintervals C and D) between frequencies 1300 and 1400 MHz can be seen in Fig. 3. The long-period spectral components (tadpole tails) of these wave trains with the characteristic period P = 63.8 s propagate faster than the medium and shortperiod ones. This characteristic period P has on average 2.3 and 1.2 wave oscillations for the tadpoles in left and right column in Fig. 3, respectively. When the duration of long-period spectral component was calculated, any portion within the COI was discarded. The average range of periods in the place of the maximal extension of the tadpole head is 36.9–70.9 and 26.0–65.5 s for tadpoles in the left and right column in Fig. 3, respectively. Some basic tadpole parameters (time subinterval C, Fig. 3) are shown in Table 2 and they are very similar at diﬀerent frequencies. Tadpoles occurring between frequencies 3980 and 4070 MHz during time subinterval A are present in Fig. 4. The long-period spectral components of these wave trains have the characteristic period P = 54.5 s with 4.5

Fig. 3. Tadpole wavelet signatures of coronal fast wave trains and their changes at individual frequencies in the range 1300–1400 MHz (time subintervals C and D). The lighter area indicates a greater power in the wavelet power spectrum and the hatched region belongs to the cone of inﬂuence (COI). The solid contour shows the 95% conﬁdence level. The ranges of powers (grey scales) are presented on right side of the spectra (common for spectra in both columns). The long-period spectral components (tadpole tails) of all wave trains have characteristic period P = 63.8 s.

wave oscillations on average. The average range of periods in the place of the maximal extension of the tadpole head is 26.7–63.1 s. These basic tadpole parameters are shown in Table 3 and they diﬀer at individual frequencies more than in the previous case (Table 2). There are some diﬀerences between individual wave trains (tadpoles) and their behavior at higher frequencies and at lower frequencies. Furthermore, the tadpoles at the same frequency but in diﬀerent time subinterval (Table 1) have some common as well as diﬀerent properties. The two tadpole series in Fig. 3 (time subintervals C and D) occur in the same frequency range 1300–1400 MHz with the same characteristic period P = 63.8 s (long-period spectral component) but with the diﬀerent wave train duration: average 2.3 wave oscillations for the earlier wave train

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Fig. 4. Tadpole wavelet signatures of coronal fast wave trains and their changes at individual frequencies in the range 3980–4070 MHz (time subinterval A). The lighter area indicates a greater power in the wavelet power spectrum and the hatched region belongs to the cone of inﬂuence (COI). The solid contour shows the 95% conﬁdence level. The ranges of powers (grey scales) are presented on right side of the spectra (common for spectra in both columns). The long-period spectral components (tadpole tails) of all wave trains have characteristic period P = 54.5 s.

Table 2 Basic tadpole parameters for the time subinterval C (left panels in Fig. 3). All wave trains have characteristic period P = 63.8 s (Duration = duration of the tadpole, Oscillations = number of wave oscillations of the period P, Range of periods = range of periods at the point of the maximal extension of the tadpole head).

Table 3 Basic tadpole parameters for the time subinterval A (see Fig. 4). All wave trains have characteristic period P = 54.5 s (Duration = duration of the tadpole, Oscillations = number of wave oscillations of the period P, Range of periods = range of periods at the point of the maximal extension of the tadpole head).

Frequency (MHz)

Duration (s)

Oscillations

Range of periods (s)

Frequency (MHz)

Duration (s)

Oscillations

Range of periods (s)

1350 1355 1360 1365 1370 1375 1380 1385 1390 1395 1400

150.6 150.0 150.0 150.0 149.4 151.3 148.7 150.0 151.4 149.4 151.8

2.4 2.3 2.3 2.3 2.3 2.4 2.3 2.3 2.4 2.3 2.4

48.9–69.3 52.9–68.5 50.0–70.1 49.5–68.5 44.7–69.3 37.7–69.3 37.7–70.9 40.8–70.9 43.2–70.1 39.5–70.1 36.9–70.9

3980 3990 4000 4010 4020 4030 4040 4050 4060 4070

205.0 187.5 281.3 217.6 250.5 206.7 188.2 235.8

3.8 3.4 5.2 3.4 4.6 3.8 3.4 4.3

26.7–59.1

234.4

4.3

32.2–58.4

(time subinterval C) and 1.2 wave oscillations for the later one (time subinterval D). Moreover, the appearance of both wave trains is diﬀerent. In the case of the earlier wave train (left column of panels in Fig. 3) we can see a full tadpole structure (tail + head) to the frequency 1370 MHz. Then, in direction to the lower frequencies, the head is fading and from the frequency 1345 MHz the whole tadpole is

38.8–63.1 27.9–61.8 32.9–62.4 47.9–54.4 31.2–59.1

decayed. In the case of the later wave trains (right column of panels in Fig. 3) we can see a full tadpole structure to the frequency 1340 MHz where the tadpole is decayed and at lower frequencies we can see only a rest of the tadpole head. The tadpole patterns of the individual wave trains (in the same time interval) at lower frequencies (Fig. 3) are very similar to each other i.e. their changes from frequency to frequency are slow.

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Diﬀerent tadpole behavior can be seen at higher frequencies (3980–4070 MHz) in Fig. 4 (time subinterval A). There is a set of diﬀerent tadpoles but with the same characteristic period P = 54.5 s (long-period spectral components) and with a similar duration (on average 4.5 wave oscillations). The process of tadpole appearance and fading (see tadpoles at frequencies 4030–4070) is signiﬁcantly more rapid and repeating. Individual tadpoles have diﬀerent heads (set of middle and short-period spectral components) and in some cases the head is absent (see tadpoles at frequencies 3990 and 4020 MHz). Sometimes, the whole tadpole is fully absent (for example at 4061 MHz). 3. Conclusions We have investigated the 22 min long decimetric type IV of radio event during the decay phase of the June 6, 2000 ﬂare observed simultaneously by the BSS and the Ondrˇejov radiospectrograph (1200–4500 MHz). For the ﬁrst time, the tadpole structures of dm-radio event are evident in their wavelet power spectra in the whole time interval and at all frequencies. We have distinguished the tadpole wavelet signatures in our observational data and it allows us to identify the corresponding waves as fast magnetoacoustic wave trains. These waves are probably trapped in a waveguide (e.g. loop) and formed by an impulsive source (e.g. ﬂare or reconnection process). We present two examples of tadpoles (Figs. 3 and 4) with diﬀerent behavior but similar characteristic period P (long-period spectral component of all wave trains) of about 60 s. The tadpoles in Fig. 3 show relatively slow changes of the same tadpole at diﬀerent frequencies. It may reﬂect that at the lower frequencies the plasma density is more smooth (no abrupt changes in density). These tadpoles have short duration of long-period spectral component (low number of wave oscillations of the characteristic period) i.e. wave damping is strong here. Furthermore, these tadpoles can decay at the lowest frequencies under study (Fig. 3). This may represent less density contrast (density ratio between external and internal plasma density of a loop) which causes that such a segment of a loop is a worse waveguide. Tadpoles at higher frequencies (Fig. 4) have longer duration of long-period spectral component (higher number of wave oscillations of their characteristic period P). On the other hand, the individual tadpoles show very rapid changes of the same tadpole at diﬀerent frequencies. Thus, individual tadpoles are diﬀerent (mainly with respect to their heads, i.e. their middle and short-period spectral components). Sometimes, we can observe only the long-period

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spectral component (tail) of the wave train that arrives as the ﬁrst at the observable point. This can represent a higher diversity in plasma density of a loop (more rapid changes of plasma density inside such a loop segment). The wave damping is strong also in this case. We have distinguished two groups of individual tadpoles with diﬀerent properties: the ﬁrst tadpole group was detected in the frequency range 1200–1600 MHz (time subintervals C and D, Fig. 3) and the second group in the frequency range 3800–4500 MHz (time subinterval A, Fig. 4). Thus, each group can belong to a diﬀerent radio emission source. If one emission source is dominant we can detect individual tadpoles with similar properties. On the other hand, the wavelet power spectra are more complex (chaotic) in the frequency range 1600–3800 MHz (i.e. we cannot see individual tadpoles). It can happen when the time series in this frequency range are inﬂuenced for example, by more than one emission source. The tadpoles can provide an evidence for MHD waves in the corona. They may provide the basics for determination of the transverse structure, density distribution and other properties of the waveguide (coronal loop). Acknowledgements H.M. acknowledges the FAPESP support for the project 2006/50039-7. H.M. and M.K. acknowledge the support from the Grant IAA300030701 of the Academy of Sciences of the Czech Republic. F.C.R.F. thanks CNPq for scholarship (proc. 310005/2005-1). J.R.C. acknowledges the CNPq support for project 475723/2004-0. J.R. acknowledges the support of the grant agency VEGA 02/6195/26. The wavelet analysis was performed using the software based on tools provided by C. Torrence and G. Compo at http:// paos.colorado.edu/research/wavelets/. References Katsiyannis, A.C., Williams, D.R., McAteer, R.T.J., et al. Eclipse observations of high-frequency oscillations in active region coronal loops. A&A 406, 709–714, 2003. Nakariakov, V.M., Arber, T.D., Ault, C.E., et al. Time signatures of impulsively generated coronal fast wave trains. Mon. Not. Roy. Astron. Soc. 349, 705–709, 2004. Roberts, B., Edwin, P.M., Benz, A.O. Fast pulsations in the solar corona. Nature 305, 688–690, 1983. Roberts, B., Edwin, P.M., Benz, A.O. On coronal oscillations. Atrophys. J. 279, 857–865, 1984. Williams, D.R., Phillips, K.J.H., Rudawy, P., et al. High-frequency oscillations in a solar active region coronal loop. Mon. Not. Roy. Astron. Soc. 326, 428–436, 2001.

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Advances in Space Research 43 (2009) 1484–1490 www.elsevier.com/locate/asr

Viewing radiation signatures of solar energetic particles in interplanetary space S.W. Kahler *, B.R. Ragot 1 Air Force Research Laboratory, RVBXS, 29 Randolph Road, Hanscom AFB, MA 01731, USA Received 18 August 2008; received in revised form 14 November 2008; accepted 6 January 2009

Abstract A current serious limitation on the studies of solar energetic particle (SEP) events is that their properties in the inner heliosphere are studied only through in situ spacecraft observations. Our understanding of spatial distributions and temporal variations of SEP events has come through statistical studies of many such events over several solar cycles. In contrast, ﬂare SEPs in the solar corona can be imaged through their radiative and collisional interactions with solar ﬁelds and particles. We suggest that the heliospheric SEPs may also interact with heliospheric particles and ﬁelds to produce signatures which can be remotely observed and imaged. A challenge with any such candidate signature is to separate it from that of ﬂare SEPs. The optimum case for imaging high-energy (E > 100 MeV) heliospheric protons may be the emission of p0-decay c-rays following proton collisions with solar wind (SW) ions. In the case of E > 1 MeV electrons, gyrosynchrotron radio emission may be the most readily detectible remote signal. In both cases we may already have observed one or two such events. Another radiative signature from nonthermal particles may be resonant transition radiation, which has likely already been observed from solar ﬂare electrons. We discuss energetic neutrons as another possible remote signature, but we rule out c-ray line and 0.511 MeV positron annihilation emission as observable signatures of heliospheric energetic ions. We are already acquiring global signatures of large inner-heliospheric SW density features and of heliosheath interactions between the SW and interstellar neutral ions. By ﬁnding an appropriate observable signature of remote heliospheric SEPs, we could supplement the in situ observations with global maps of energetic SEP events to provide a comprehensive view of SEP events. Published by Elsevier Ltd. on behalf of COSPAR. Keywords: Solar energetic particles; Interplanetary magnetic ﬁelds; Coronal mass ejections

1. Introduction Forecasting the occurrence of SEP events has become increasingly important as we consider their impact on the human exploration of space (Turner, 2006). At the current time we must rely on solar ﬂare and coronal mass ejection (CME) signatures to predict the temporal, spatial, and energetic variations of heliospheric SEP events, but the presumed SEP production in CME-driven shocks can be only loosely connected to those solar signatures (Kahler, 2001). Furthermore, we rely on statistical studies of in situ observations to *

1

Corresponding author. E-mail address: [email protected] (S.W. Kahler). NRC Senior Research Associate.

0273-1177/$36.00 Published by Elsevier Ltd. on behalf of COSPAR. doi:10.1016/j.asr.2009.01.013

determine the characteristics of the heliospheric SEP events. Although the 1 AU in situ observations give us detailed information on heliospheric SEP spectra and composition, the lack of a complementary global context for SEP production and the loss of SEP source information imposed by particle scattering on magnetic ﬂuctuations during SEP transport in the inner heliosphere are clearly severe impediments for our characterization and understanding of those SEP events. The low ambient densities and weak magnetic ﬁelds of the heliosphere restrict any radiative SEP signatures to low levels not yet observed. In contrast to these limitations, the SEPs accelerated in solar coronal ﬂare structures are remotely observed and diagnosed with microwave, optical, X-ray, and c-ray emission and neutron detections. The RHESSI spacecraft is a solar observatory that provides

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dedicated solar ﬂare observations in the X-ray and c-ray range up to 20 MeV with good spatial, temporal and spectral resolution (Lin et al., 2002). Here we consider some possible remote signatures of heliospheric SEPs that might serve as future observable diagnostics. Our recent calculations of c-ray line and continuum emissions from SEP interactions with solar wind (SW) ions (Kahler and Ragot, 2008) are brieﬂy reviewed. We also examine several other SEP interactions that might serve to produce observable heliospheric SEP signatures. It is important that we consider all possible forms of remote SEP information – radiative, magnetic, electric, particle, or other. We point out that at the relatively low energies of SW particles, remote observations, perhaps not thought possible several decades ago, are now becoming a reality and will complement various in situ observations in the heliosphere and at the termination shock and heliosheath.

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We have recently considered (Kahler and Ragot, 2008) the possibility of remote detection of c-rays produced in the near-Sun heliosphere by the interaction of very intense SEPs with SW particles, as shown in Fig. 1. This work was motivated by three factors. First, in the galaxy c-ray line (Tatischeﬀ and Kiener, 2004) and high-energy (E > 70 MeV) continuum emission (Strong and Mattox, 1996) resulting from cosmic ray collisions with interstellar gas and dust has been observed. Second, SEP spectra and temporal variations for relatively large gradual events have been well characterized during the past solar cycle (Mewaldt et al., 2005), providing a SEP data base for selecting events to model in the near-Sun heliosphere. Finally, the GLAST

spacecraft (Bhattacharjee, 2008), launched on 2008 June 11, will provide a new capability for sensitive measurements of c-rays from heliospheric SEPs. Two c-ray regimes were explored (Kahler and Ragot, 2008). First, we considered c-ray line emission resulting from excitation by 3–30 MeV nuc1 ions on SW gas and dust in the 5–15 R region using the estimated peak 3–30 MeV proton spectrum of the large gradual SEP event of 2003 October 28 and the excitation cross sections of Kozlovsky et al. (2002). The calculated intensities for the strongest lines (1.37, 4.44, and 6.13 MeV of 24Mg, 12C, and 16O, respectively) were slightly lower than the combined observed diﬀuse extragalactic component background (Strong et al., 1996) and the weaker calculated background from the inverse Compton scattering of cosmic ray electrons on the solar photon halo (Moskalenko et al., 2006). Intensities consistent with the inverse Compton scattered component have recently been detected in the two higher energy ranges of 100–300 MeV and >300 MeV by Orlando and Strong (2008). However, over the short (1–10 h) timescale of the peak of a gradual SEP event less than a single count would be recorded in the GLAST Burst Monitor, rendering this approach hopeless. The prospects for an event detection were more favorable, however, for p0-decay c-rays resulting from collisions of high energy (E J 300 MeV) SEP protons on SW ions. We used an E2 diﬀerential energy spectrum to represent the 2005 January 20 SEP event peak (Fig. 2) and calculated not only that the p0-decay c-ray intensity was three orders of magnitude above background but that over the event peak hour the Large Area Telescope (LAT) on GLAST would have detected J 104 counts (using the correct 0.6 cm2 s1 sr1 value for the c-ray intensity of I p0 , which we erroneously took as 0.3 cm2 s1 sr1). The SEP diﬀerential ﬂuence spectrum for the January 20 event has also

Fig. 1. c-Ray imaging of both ﬂare (dashed-dot line and wiggly arrow) and interplanetary (thin dashed lines and wiggly arrows) SEP populations. The CME drives a shock (thick dashed line) from which SEPs propagate outward (straight arrows) to interact with solar wind and dust ions (gray area) and produce c-rays. From Kahler and Ragot (2008).

Fig. 2. Time proﬁles of the E > 100 MeV intensities of the largest SEP events of the last 30 years. The 2005 January 20 event used in our c-ray calculation had the fastest rise of any of the events. From Mewaldt et al. (2005).

2. Candidate SEP signatures 2.1. c-Ray continuum emission

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been calculated from neutron monitor data and is steeper and less intense than the intensity spectrum we used. When the calculated neutron-monitor integral ﬂuence above 300 MeV matches that of our normalized intensity spectrum, the c-ray ﬂux and total LAT counts are reduced by a factor of 3 (R. Murphy and A. Tylka, private communication), but still produce a very detectible LAT signal. Details of the line and p0-decay emission calculations are given in Kahler and Ragot (2008). A major problem in detecting any near-Sun heliospheric SEP event signature is to distinguish that signal from a similar signature of solar ﬂare SEPs. The low signal intensities and the very high (610 4 R) detector angular resolution required to make that distinction combine to make a formidable observing challenge. We (Kahler and Ragot, 2008) suggested that the ﬂare and heliospheric SEP signals could be distinguished from each other either temporally, when a later heliospheric signal follows or dominates an earlier ﬂare signal, or spatially, when the ﬂare region lies over the solar limb and only the heliospheric component is observed. Ryan (2000) considered in detail how protons accelerated in antisunward propagating shocks might precipitate back to the lower corona or chromosphere as an explanation for observed long-duration solar c-ray ﬂares. His concept is similar to our suggestion of a temporal separation between ﬂare and heliospheric SEP sources, except that he would have the later collisions and c-ray production from the heliospheric SEPs occurring in much lower and denser coronal regions rather than in the SW. 2.2. Positron-decay 0.511 MeV line emission We expect that a corresponding heliospheric SEP signature resulting from collisions of SEP ions with SW ions should match each observed signature of ﬂare SEP ions with solar ambient atmospheric ions. Collisions of high energy (E J 300 MeV) SEP protons on SW ions will produce not only the p0-decay c-rays considered in our earlier work (Kahler and Ragot, 2008) but also p+ and p mesons, which decay into l+ and l mesons and then into positrons and electrons. Positrons are also produced as products of collisions of lower energy (E J 10 MeV) SEPs with ambient ions to produce radioactive b+-emitting nuclei (Ramaty et al., 1975). Production of those nuclei can be greatly enhanced if the SEP composition has a high 3He abundance in the 1–10 MeV nuc1 energy range (Kozlovsky et al., 2004). The positrons decay into a pair of 0.511 MeV c-rays when they annihilate directly with ambient electrons or after forming positronium. From positronium only 25% decay into the 0.511 MeV c-rays, while the remaining 75% undergo a three-c-ray decay. Could the 0.511 MeV line also be observed from the heliospheric SEP-generated positrons as it is in some solar ﬂares? The p+-decay positrons are formed with energies of 30 MeV and for annihilation they require thermalization times of 1013/n1 s, where n1 is the ambient density in cm3 (Ramaty and Murphy, 1987). In heliospheric regions of

density n < 108 cm3 those relativistic positrons will leave the inner heliosphere with essentially no 0.511 MeV line emission. The b+-emission positrons have lower characteristic energies of 1 MeV and shorter thermalization times of 4 1012/n1 s, but they too should rapidly propagate away from the inner heliosphere before decay into the 0.511 MeV line is possible. Thus, the long positron thermalization times preclude the possibility of observing heliospheric 0.511 MeV line emission. Direct detection of the positrons themselves at 1 AU is not precluded here, but as charged particles scattered by magnetic ﬁeld ﬂuctations, they lose the spatial information of their sources that we are seeking. 2.3. Neutrons and neutron-capture 2.23 MeV c-ray emission Since energetic neutrons travel directly from their source regions to an observer, directional information on neutrons produced by heliospheric SEP collisions with SW ions could produce information on the SEP temporal–spatial distribution. Neutrons are produced primarily by p–a interactions above 30 MeV nuc1, where the production cross sections are only slightly energy dependent (Ramaty et al., 1975). In the case of heliospheric SEPs the thin-target model of interactions is appropriate for neutron production calculations, rather than the more eﬃcient thick-target model usually assumed (i.e., Murphy et al., 1987) for ﬂare SEPs, which are stopped in the dense ﬂare target region. The b decay of the neutrons en route to the observer will modify the observed neutron energy spectrum, but this could provide model-dependent information on the SEPsource distance from the observer that would not be available from the heliospheric p0-decay c-rays. Neutron emission in the 36–100 MeV range associated with a solar ﬂare located 60–90 behind the solar east limb was observed on 1991 June 1 (Murphy et al., 1999). Comparison of that event with a second neutron and c-ray ﬂare event on 1991 June 4 indicated that a thin-target neutron source was possible only if the SEP spectrum was extremely hard. Although the June 1 ﬂare was an extremely energetic event (Kane et al., 1995), it suggests that energetic neutron emission could, with a suﬃciently sensitive detector, serve as a remote signature of heliospheric SEP events. The energetic neutrons produced by p–a collisions between energetic heliospheric SEPs and the SW ions will produce 2.23 MeV c-ray line emission from capture by SW protons. However, two neutron timescales limit this capture process: the 15-min lifetime against b decay and the required thermalization by elastic scattering before SW proton capture. In solar ﬂares the 2.23 MeV emission is generated in the n > 1016 cm3 density region of the photosphere after a thermalization time of 100 s for 1–100 MeV neutrons (Wang and Ramaty, 1974; Ramaty et al., 1975). However, the SW densities, lower by orders of magnitude, preclude any neutron thermalization, and detectible 2.23 MeV line emission from heliospheric SEPs is therefore not expected. We note, however, that

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2.23 MeV emission was observed on 1989 September 29 in association with a GLE, a fast CME, and a solar ﬂare located 50–150 behind the solar west limb. Vestrand and Forrest (1993) and Cliver et al. (1993) suggested that a fraction of the E > 30 MeV protons at the coronal shock may have precipitated back to the solar atmosphere and then produced the observed emission via neutron generation. Our consideration here of neutron production by heliospheric SEPs suggests the alternative possibility of direct atmospheric precipitation of the energetic neutrons from the SEP–SW interactions, which, although unlikely, avoids the problem of achieving charged particle precipitation through converging coronal magnetic ﬁelds. 2.4. Electron synchrotron emission Bursts of gyrosynchrotron radiation from E J 100 keV electrons in solar ﬂare loops are commonly observed in the microwave range (Bastian et al., 1998). Can synchrotron emission also be observed from transient energetic electron populations that escape the Sun? Bastian et al. (2001) described a fast CME on 1998 April 20 in which the expanding CME loops were imaged directly in radio wavelengths with the Nancay radioheliograph out to 3.5 R. Their interpretation was that the emission was synchrotron emission from nonthermal electrons with energies from 0.5 to 5 MeV in ﬁelds of 0.1 to several G. The longduration phase of an X-class ﬂare event on 2003 November 3 was also interpreted in terms of electron gyrosynchrotron emission from a large coronal structure with B 2 G (Dauphin et al., 2005). Matching modulations were observed in hard X-rays by RHESSI and in the decimetric/metric continuum at the Nancay Radioheliograph during a fast CME. Another event, in which radio loops were observed in a CME out to >2.1 R on 2001 April 15 (Maia et al., 2007), provides a third example. The inferred electron high-energy cutoﬀ ranged from 1 to 10 MeV and the estimated ﬁeld B was 1 G. These events were imaged at the four Nancay observing frequencies from 432 to 164 MHz, and their cospatial emission at diﬀerent frequencies indicated synchrotron, not plasma emission, as the source. The 1998 and 2001 events originated from the southwest solar limb and were accompanied at 1 AU by large E J 100 keV electron and E > 10 MeV proton events, suggesting that the escaping nonthermal electrons may have been imaged near the Sun. Can we see synchrotron emission from electrons propagating even farther from the Sun? An interplanetary type-II-like burst was observed with the Wind/WAVES instrument on 2003 June 17–18, which Bastian (2007) interpreted as synchrotron emission. The burst followed a strong type III burst and preceded a type II burst and was observed when the CME height was J 10 R (Fig. 3). He argued that the frequency band width, drift rates and the single smoothly varying emission lane of the burst were inconsistent with plasma emission from a propagating shock. Acceptance of a new class of low-frequency (<5 MHz) syn-

Fig. 3. Bottom panel: example of IP type II-s and type II-p radio bursts on 2003 June 17 observed with the Wind/WAVES RAD1 and RAD2 receivers. The type II-s burst follows the type III fast-drift bursts (not marked) and has a single smooth, diﬀuse, broad-band proﬁle compared with the subsequent type II-p burst. Top panel: The GOES-10 class M6.8 soft X-ray ﬂare proﬁle and the height of the leading edge of the CME observed on the southeast quadrant of the sun with the SOHO/LASCO. From Bastian (2007).

chrotron emission must await the discovery of further such examples, but, if found, these bursts could be the basis for tracking distributions of E J 100 keV electrons in space. Deriving detailed spatial information about electron and shock sources would require direction ﬁnding and triangulation analyses from space observations such as the STEREO/SWAVES instruments (Bougeret et al., 2008). A more ambitious proposal is the Solar Imaging Radio Array, a mission to perform aperture synthesis imaging of low frequency (<15 MHz) bursts (MacDowall et al., 2005). A large number of baselines would be provided by a 12–16 microsatellite constellation in Earth orbit about 0.5 106 km from the Earth. High-resolution images of the entire sky over a range of frequencies should go far to determine emission mechanisms of solar bursts and the roles of SEPs in producing those bursts (see Fig. 4). 2.5. Resonant transition radiation Resonant transition radiation (RTR) is an incoherent continuum emission mechanism in astrophysical plasmas. Transition radiation arises when nonthermal charged particles pass through small-scale inhomogeneities, but it was considered to be weak until Platonov and Fleishman (2002) showed that its intensity was greatly enhanced by plasma resonance (hence RTR) at frequencies just above

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Fig. 4. Multiple ﬁbers and striae observed in the 1–10 MHz region by the Wind/WAVES instrument on 1997 December 12. In this unusual case the zebra pattern is observed both in the continuum and in fast-drift type III bursts. This is Fig. 1 of Chernov et al. (2007) with kind permission of Springer Science and Business Media.

the local plasma frequency xp (Nita et al., 2005). The emission intensity scales as h(Dn)2i/n2 (where h(Dn)2i is the mean square of the inhomogeneities, so the emission is enhanced in more turbulent regions (Fleishman et al., 2005). Evidence for RTR has been found in solar ﬂare bursts characterized by cospatial and simultaneous emission in the decimetric range from RTR and in the centimetric range from gyrosynchrotron emission (Fleishman et al., 2005; Nita et al., 2005). The interpretation is that the lower energy (6100 keV) electrons produce the RTR, and the higher energy ( J 300 keV) electrons the gyrosynchrotron emission. RTR is suppressed at x < x2p =xB , where xB is the electron gyrofrequency, so dense, high-b plasmas are favorable for enhanced RTR. The calculated turbulence levels in several ﬂare studies (Nita et al., 2005; Fleishman et al., 2005; Yasnov et al., 2008) are h(Dn)2i/n2 107– 105. Can RTR also be observed outside the ﬂaring active regions? Chernov et al. (2007) have discussed ﬁne structure observed in the 14–5 MHz observing range of the Wind/ WAVES radio receiver. In 14 selected events the ﬁne structure was in the form of ﬁbers, similar to the zebra patterns observed in the metric range. In most of the events shock fronts were observed in white light coronagraph images to cross narrow streamers in the wakes of CMEs. Their suggestion was that fast electrons at the shock front traversed the density inhomogeneities of the streamers to produce RTR. They point out that while Dn/n may be only a few percent in the corona, it can exceed unity in the CME streamers. The interplanetary scintillation (IPS) observations of a CME at about 10 R (Lynch et al., 2002) indicated that Dn/n was somewhat lower than in the preceding slow solar wind, but still on the order of unity.

An 10 min enhancement in the IPS signal implied a small structure 30 times denser than the background. These observations support the point of Chernov et al. (2007) that the large density ﬂuctuations of the near-Sun region may provide a favorable situation for observing RTR. 2.6. EUV spectroscopy While the above techniques detect direct products of heliospheric SEPs, EUV spectroscopy can provide diagnostics of shock regions in which SEPs may be produced. Remote observations of various ionic line proﬁles from shock regions can yield their suprathermal velocity distributions as well as abundance and ionic charge-state information as functions of space and time (Pagano et al., 2008). Suprathermal protons might be detected from Lya emission following charge exchange with SW hydrogen atoms (Kahler et al., 1999). Line broadening attributed to shock heating has already been detected at the fronts of a number of CMEs (Ciaravella et al., 2006) with SOHO/UVCS observations, but a detector with much higher sensitivity and wider spectral range would be needed to get line proﬁles of many ions with various charge-to-mass ratios. 3. Imaging solar wind structures In the preceding sections we have suggested potential techniques for imaging various kinds of radiations produced by the energetic electrons or ions escaping the corona to become heliospheric SEP events. One or more of these techniques could prove to be a valuable complement to in situ SEP observations for deﬁning the heliospheric spatial and temporal variations of SEP events. In general,

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the proposed observations would require high detector sensitivities and angular resolutions to separate the transient SEP radiant signals from high backgrounds. Here we point out that imaging structures in the very tenuous solar wind (SW) has also been a challenge for heliospheric studies. Charge exchange between SW a particles and interstellar H atoms in the heliosphere produces emission in the 30.4-nm line. Because the charge-exchange process is velocity dependent, 30.4-nm spectral observations could determine spatial variations of SW speeds and densities, but the expected weak radiance (Gruntman et al., 2006) would limit realistic observations to several all-sky images per year. A recent proposal by Morgan et al. (2008) is to measure the 103.196 and 103.76 nm line proﬁles of the collisional component of O VI in the SW along lines of sight directed away from the Sun. On an inner-heliospheric spacecraft the O+5 bulk ﬂows and eﬀective ionic temperatures deduced from those EUV lines could complement the in situ SW particle measurements to determine extended SW structures. Thomson-scattering of solar white-light photons was also at one time considered a diﬃcult challenge. However, because of a very steady zodiacal whitelight background the Solar Mass Ejection Imager has imaged CMEs (Webb et al., 2006) two orders of magnitude fainter than that background (Jackson et al., 2004). Recently, images of SW density streams have been obtained by heliospheric imagers on the STEREO spacecraft (Sheeley et al., 2008; Rouillard et al., 2008). The energetic neutral atoms (ENAs) produced by interactions of SW ions with interstellar neutrals are another important kind of remote SW observations. Model simulations have shown that 25–100 keV ENA ﬂuxes produced from charge exchange of ions accelerated at corotating interaction regions could explain ENA ﬂuxes observed on the SOHO spacecraft (Ko´ta et al., 2001). The recent Voyager 2 in situ plasma and nonthermal ion measurements (Decker et al., 2008) and STEREO directional measurements of 4–20 keV ENAs from the heliosheath (Wang et al., 2008) have shown the importance of nonthermal pick-up ions to the dynamical processes of the termination shock. The Interstellar Boundary Explorer (IBEX) mission (McComas et al., 2006), successfully launched on 2008 October 19, will map the full sky in ENAs in the energy range from 10 eV to 6 keV. McComas et al. (2004) have stressed the important complementarity of the IBEX global maps to the Voyager 1 and 2 in situ measurements at the termination shock and heliosheath to provide a full picture of the SW interaction with the interstellar medium. The large-scale structure of the inner heliospheric magnetic ﬁeld is also beginning to be explored. Magnetic ﬁeld characteristics of large ﬂux-rope CMEs have been determined from Faraday rotation measurements of single-frequency signals from spacecraft (Jensen and Russell, 2008). A much more detailed determination of both transient and slowly evolving SW magnetic ﬁelds is anticipated from Faraday-rotation observations with the Mileura Wideﬁeld Array (Salah et al., 2005) operating at 80–300 MHz.

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The mapping of heliosheath ENAs and SW magnetic ﬁelds and the imaging of white-light SW streams and ICMEs are cases in which remote signatures of dynamic SW processes have been identiﬁed and used as the basis of global measurements to complement the in situ measurements. As with the SW and heliosheath, the ﬁrst studies of SEPs have been limited to in situ observations. The successes in probing large-scale SW and heliosheath features with remote observations should encourage us to seek similar remote signatures of SEPs to complement the extensive current in situ SEP measurements. 4. Summary Our goal here has been to encourage the development of concepts for the remote observation of heliospheric SEPs. We have shown that remote observations are becoming a reality for SW and heliosheath structures. These remote observations add the necessary context to interpret properly and understand the physics deduced from the in situ observations. We have suggested a number of candidate remote signatures produced by various kinds of heliospheric SEPs interacting with SW ﬁelds or particles, which we summarize as: 1. Pion-decay c-rays as signatures of E J 300 MeV nuc1 ions, 2. Energetic neutrons as signatures of E J 30 MeV nuc1 ions, 3. Electron synchrotron emission as a signature of E J 0.3 MeV electrons, and 4. Resonant transition radiation (RTR) as a signature of E [ 100 keV electrons and of energetic ions. We have also considered but found that 4–7 MeV ion deexcitation, the 2.23 MeV neutron-capture, and 0.511 MeV positron annihilation c-ray lines from heliospheric SEPs would be too weak for observation. There may well be other possibilities that remain to be discovered. Acknowledgements The authors thank the two reviewers for their very helpful comments on the manuscript. References Bastian, T.S. Synchrotron radio emission from a fast halo coronal mass ejection. Astrophys. J. 665, 805–812, 2007. Bastian, T.S., Benz, A.O., Gary, D.E. Radio emission from solar ﬂares. Ann. Rev. Astron. Astrophys. 36, 131–188, 1998. Bastian, T.S., Pick, M., Kerdraon, A., Maia, D., Vourlidas, A. The coronal mass ejection of 1998 April 20: direct imaging at radio wavelengths. Astrophys. J. 558, L65–L69, 2001. Bhattacharjee, Y. GLAST mission prepares to explore the extremes of cosmic violence. Science 320, 1008–1009, 2008. Bougeret, J.-L. et al. S/WAVES: the radio and plasma wave investigation on the STEREO mission. Space Sci. Rev. 136, 487–528, 2008.

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Advances in Space Research 43 (2009) 1491–1508 www.elsevier.com/locate/asr

Dynamical evolution of high area-to-mass ratio debris released into GPS orbits L. Anselmo *, C. Pardini Space Flight Dynamics Laboratory, ISTI/CNR, Via G. Moruzzi 1, 56124 Pisa, Italy Received 16 October 2008; received in revised form 19 January 2009; accepted 20 January 2009

Abstract A large set of simulations, including all the relevant perturbations, was carried out to investigate the long-term dynamical evolution of ﬁctitious high area-to-mass ratio (A/M) objects released, with a negligible velocity variation, in each of the six orbital planes used by Global Positioning System (GPS) satellites. As with similar objects discovered in near synchronous trajectories, long lifetime orbits, with mean motions of about 2 revolutions per day, were found possible for debris characterized by extremely high area-to-mass ratios. Often the lifetime exceeds 100 years up to A/M 45 m2/kg, decreasing rapidly to a few months above such a threshold. However, the details of the evolution, which are conditioned by the complex interplay of solar radiation pressure and geopotential plus luni-solar resonances, depend on the initial conditions. Diﬀerent behaviors are thus possible. In any case, objects like those discovered in synchronous orbits, with A/M as high as 20–40 m2/kg, could also survive in this orbital regime, with semi-major axes close to the semi-synchronous values, with maximum eccentricities between 0.3 and 0.7, and with signiﬁcant orbit pole precessions (faster and wider for increasing values of A/M), leading to inclinations between 30° and more than 90°. Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: High area-to-mass ratio objects; GPS semi-synchronous orbits; Orbit stability and long-term dynamical evolution; Orbital debris; Solar radiation pressure; Luni-solar and geopotential resonances

1. Introduction Optical observations have led to the discovery of a population of faint uncataloged objects with mean motions of about 1 revolution per day and orbital eccentricities as high as 0.6 (Schildknecht et al., 2004, 2008; Agapov et al., 2005). This population may be explained by objects with very high area-to-mass ratios (A/M), up to 30 m2/kg, released with a negligible velocity change (DV) in geostationary or low eccentricity geosynchronous orbits (Liou and Weaver, 2004, 2005; Anselmo and Pardini, 2005, 2007; Chao, 2006; Pardini and Anselmo, 2008; Valk et al., 2008; Valk, 2008). In fact, direct solar radiation pressure may signiﬁcantly

*

Corresponding author. Tel.: +39 050 315 2952; fax: +39 050 315 2040. E-mail addresses: [email protected] (L. Anselmo), luciano. [email protected], [email protected] (C. Pardini).

aﬀect the eccentricity of these high A/M objects, leaving their semi-major axis or mean motion basically unaltered. The long-term orbit evolution of synchronous (1:1 resonance with Earth rotation) objects characterized by very high area-to-mass ratios has already been analyzed in detail in the above-mentioned papers. The purpose of this study was, instead, to investigate the evolution over 100 (and in some cases 200) years of semi-synchronous objects, with 0.05 m2/kg 6 A/M 6 100 m2/kg, released with negligible DV in the orbits used by the Block II satellites of the Global Positioning System (GPS). The trajectories were propagated with the latest version of the special perturbations numerical code SATRAP (Pardini and Anselmo, 1994; Anselmo and Pardini, 2007), taking into account geopotential harmonics up to the 16th degree and order, lunisolar attraction and direct solar radiation pressure, including the eclipses due to the Earth’s shadow as described in Pardini and Anselmo (2008). The numerical integration

0273-1177/$36.00 Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2009.01.017

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of the equations of motion was carried out with a Runge– Kutta single step method with variable step-size control (Kwok, 1987). For induced high eccentricity orbits with a perigee altitude of below 1000 km, the perturbing eﬀects of air drag were also considered, using the 1976 United States Standard Atmosphere. The radiation pressure coefﬁcient CR was set to 1.2, and the drag coeﬃcient CD to 2.2, when applicable. In the simulations, ﬁctitious objects with varying A/M were released, with a negligible DV, in each of the six orbital planes used by the GPS constellation. The orbital state vectors of six operational satellites, as at 16–17 April 2007, were chosen as initial conditions (Table 1). For reasons that will be explained in Section 2, together with the nomenclature used, the orbits of the following Block II satellites, one for each constellation plane, were selected: GPS-44 (A3), GPS-51 (B1), GPS-57 (C4), GPS-56 (D1), GPS-53 (E2), and GPS-55 (F4). 2. Properties of the GPS Block II orbits The nearly circular orbits of GPS Block II satellites, with periods equal to 1/2 of a sidereal day (718 min) and semi-major axes close to 26,560 km, are in deep 2:1 resonance with the Earth’s rotation. The satellites are distributed in six orbital planes with an inclination of about 55° with respect to the equator. The ascending nodes of the six planes are separated by 60° in right ascension. Each plane is identiﬁed by a capital letter, from A to F, while the satellites in each plane are identiﬁed by a number. Due to the GPS orbit ground track geometry and repeat frequency, it is not diﬃcult to conclude that the most important geopotential resonant term aﬀecting the semimajor axis is in this case J32 (instead of J22, as with geosynchronous orbits), followed by J44, J22 and J52 (Hugentobler, 1998; Ineichen et al., 2003; Beutler, 2005). The resonance outcome, due mainly to J32, is in this case the existence of stable (28°E, 208°E) and unstable (118°E, 298°E) longitudes of the ascending nodes kN, leading to a nearly harmonic motion around the stable points in the Poincare´ phase plane (kN, a) deﬁned by the longitude of the ascending node kN and the semi-major axis a (Hugentobler, 1998; Ely, 1999). Another property of the GPS Block II orbits is their closeness to the ‘‘critical” inclination value 54.74° (Beutler, 2005). For such inclination, and for its supplementary

(125.26°), the mean motion in the potential of an oblate Earth, averaged over a revolution, is the same as in the potential of a spherical Earth. The orbital inclination of GPS Block II satellites is also close to signiﬁcant luni-solar resonances. The resonant inclination at 56.1° is associated with luni-solar harmonic terms with an angular argument of (2x + X) (Hughes, 1980; Ely, 1999; see also Rossi, 2008, for a recent review), while the resonant inclination at 53.3° is associated with a lunar harmonic term with an angular argument of (2x + XL p) (Ely, 1999), where x is the argument of perigee, X is the right ascension of the ascending node and XL is the longitude of the ascending node of the lunar orbit with respect to the ecliptic plane. 3. Long-term evolution for small values of the area-to-mass ratio As mentioned at the end of Section 1, the simulations described in this paper were carried out assuming the orbital state vectors of six operational satellites, as at 16–17 April 2007, as initial conditions. This was done to minimize the number of computer runs as much as possible (to about 180, taking into account the various A/M ratios, force model options and sensitivity analyses), ensuring at the same time the maximum generality of the results. Therefore, the satellite’s initial conditions (Table 1) were chosen not only to cover all the six constellation orbital planes (with one satellite per plane), but also to guarantee the appropriate spread in the semi-major axis, resonant argument (2x + X), inclination, and the longitude of ascending node (see Figs. 1–3). For small values of the area-to-mass ratios, such as those typical of spacecraft and upper stages, the long-term dynamical evolution of objects abandoned in GPS Block II orbits is dominated by geopotential and luni-solar perturbations. Fig. 4 clearly shows the anti-clockwise nearly pendular motion around the stable points in the Poincare´ phase plane (kN, a), mainly driven by the J32 resonant harmonic. The period of small amplitude motion (satellite E2) is about 8.4 years, increasing to about 14 years far away from the stable equilibrium points (satellite B1). This nearly pendular motion explains the behavior of the semi-major axis, shown in Fig. 5 (over about 30 years) and Fig. 6 (over 100 years). The amplitude and the period of the main oscillation very much depend on the initial

Table 1 Initial conditions (mean elements) used for the simulations. Satellite orbit GPS-44 GPS-51 GPS-57 GPS-56 GPS-53 GPS-55

Epoch (UTC) (A3) (B1) (C4) (D1) (E2) (F4)

17 16 16 17 16 16

Apr Apr Apr Apr Apr Apr

2007 2007 2007 2007 2007 2007

03:05:55.335 00:46:42.433 07:29:14.720 11:03:07.094 17:28:58.794 23:06:57.974

Semi-major axis (km)

Eccentricity

Inclination (°)

Right ascension of ascending node (°)

Argument of perigee (°)

Mean anomaly (°)

26,560.826 26,559.890 26,559.616 26,559.355 26,560.432 26,560.259

0.0096227 0.0041632 0.0026456 0.0087123 0.0048672 0.0046480

56.0150 55.2000 54.9919 54.2489 54.5093 55.6068

72.5607 130.7562 190.2977 249.6308 312.7360 9.9670

159.9943 315.4985 189.3242 130.4077 265.1898 150.7686

200.3662 44.2377 170.6826 230.3125 94.2809 209.5237

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Fig. 1. Distribution of orbit planes and semi-major axes of the GPS operational satellites (blue circles), as of 17 April 2007. The satellites whose orbits were chosen for the simulations presented in this paper are explicitly labeled.

Fig. 2. Distribution of luni-solar resonant arguments (2x + X) and inclinations of the GPS operational satellites (blue circles), as of 17 April 2007. The satellites whose orbits were chosen for the simulations presented in this paper are explicitly labeled.

‘‘distance” from the stable equilibrium points in the plane (kN, a) and are basically determined by the dominant resonant geopotential harmonics, above all J32 and, to a lesser extent, J44, J22 and J52. However, in the long-term, luni-

solar eﬀects may signiﬁcantly disturb the nearly pendular oscillation of the semi-major axis. This can be seen in Fig. 7, where the impact of the various perturbations is shown for satellite A3, over a time span of 200 years.

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Fig. 3. Distribution, as of 17 April 2007, of the chosen satellite orbits in the Poincare´ phase plane deﬁned by the longitude of the ascending node and the semi-major axis.

Fig. 4. Near pendular evolution of the chosen satellite orbits in the Poincare´ phase plane deﬁned by the longitude of the ascending node and the semimajor axis. The area-to-mass ratios were set to 0.05 m2/kg and all perturbations described in Section 1 were included.

The increase in eccentricity is also strongly aﬀected by the initial conditions (Figs. 8 and 9) and is mostly driven by the interaction of J2, which is the main term of the gravitational potential resulting from the oblate Earth, with luni-solar eﬀects due to inclination resonances (Fig. 10).

The relatively small (for A/M = 0.05 m2/kg) oscillation induced by solar radiation pressure, with a period of nearly one year, is also clearly noticeable, particularly in Fig. 8. The interaction of J2 with luni-solar perturbations also dominates the inclination evolution (Figs. 11 and 12).

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Fig. 5. Mean semi-major axis evolution of the chosen satellite orbits over about 30 years. The area-to-mass ratios were set to 0.05 m2/kg and all perturbations described in Section 1 were included.

Fig. 6. Mean semi-major axis evolution of the chosen satellite orbits over 100 years. The area-to-mass ratios were set to 0.05 m2/kg and all perturbations described in Section 1 were included.

Disregarding minor shorter term eﬀects due to geopotential harmonics, Moon and Sun attraction and direct solar radiation pressure, a relatively wide oscillation, with the period of the orbit nodal regression (26 years) and with

a phase depending on the initial value of the right ascension of the ascending node, is superimposed on a longer period luni-solar trend of even greater amplitude (Fig. 12).

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Fig. 7. Eﬀects of the perturbations on the mean semi-major axis evolution for satellite A3. The time span considered was 200 years and the area-to-mass ratio was set to 0.05 m2/kg. The perturbations included in the analysis were the geopotential harmonics up to the 16th degree and order (16 16), the lunar (Moon) and solar (Sun) third body attraction and direct solar radiation pressure with eclipses (SRP). The diﬀerences between the (16 16) and (16 16 + SRP) results are very small and the two curves practically overlap.

Fig. 8. Mean eccentricity evolution of the chosen satellite orbits over 100 years. The area-to-mass ratios were set to 0.05 m2/kg and all perturbations described in Section 1 were included. The relatively small oscillation induced by solar radiation pressure, with a period of nearly one year, is clearly noticeable.

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Fig. 9. Anti-clockwise precession of the mean eccentricity vector (ecos x, esin x) for the chosen satellite orbits over 100 years. The area-to-mass ratios were set to 0.05 m2/kg and all perturbations described in Section 1 were included.

Fig. 10. Eﬀects of the perturbations on the mean eccentricity evolution for satellite A3. The time span considered was 200 years and the area-to-mass ratio was set to 0.05 m2/kg. The perturbations included in the analysis were the geopotential harmonics up to the 16th degree and order (16 16), the lunar (Moon) and solar (Sun) third body attraction and direct solar radiation pressure with eclipses (SRP). The diﬀerences between the (16 16) and (16 16 + SRP) results are very small and the two curves practically overlap, except for a slight oscillation caused by solar radiation pressure, with a period of nearly 1 year.

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Fig. 11. Mean inclination evolution of the chosen satellite orbits over 100 years. The area-to-mass ratios were set to 0.05 m2/kg and all perturbations described in Section 1 were included.

Fig. 12. Eﬀects of the perturbations on the mean inclination evolution for satellite A3. The time span considered was 200 years and the area-to-mass ratio was set to 0.05 m2/kg. The perturbations included in the analysis were the geopotential harmonics up to the 16th degree and order (16 16), the lunar (Moon) and solar (Sun) third body attraction and direct solar radiation pressure with eclipses (SRP). The diﬀerences between the (16 16) and (16 16 + SRP) results are very small and the two curves practically overlap, except for a slight oscillation induced by solar radiation pressure, with a period of nearly 1 year.

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4. Long-term evolution for high values of the area-to-mass ratio Concerning the orbital lifetime of objects with area-tomass ratios of up to 100 m2/kg, released with negligible DV in each of the six orbit planes used by GPS Block II satellites, the results obtained are summarized in Fig. 13. All the objects analyzed with A/M of up to 23 m2/kg showed a lifetime greater than 100 years, with the semi-major axis and orbital period remaining close to the semi-synchronous values, as will be shown below. Moreover, all the objects with A/M of up to 45 m2/kg exhibited a lifetime greater than 35 years (often greater than 100 years), again with the semi-major axis and orbital period remaining close to the semi-synchronous values. However, for 45 m2/kg < A/M < 80 m2/kg, depending on the initial conditions, the eccentricity became so large, and the perigee altitude so low, that an orbital decay occurred in a few months. But even in these cases, the semi-major axis and the orbital period remained close to the semi-synchronous values until reentry. Figs. 4 and 13 highlight that the orbits that were most prone to instability for A/M 6 45 m2/kg were those (B1 and D1) farthest from the stable equilibrium points in the Poincare´ phase plane (kN, a). Nevertheless, for A/M > 45 m2/kg, the most stable orbits were those in an intermediate position, that is in between the stable and unstable equilibrium points (C4, D1 and F4, respectively). Regarding the long-term evolution of the semi-major axis, eccentricity and inclination, the following ﬁgures show the results obtained with the B1 initial conditions. This is only a small subset of the overall results, but for the sake of brevity just one set of examples was chosen

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to clarify the most signiﬁcant aspects of the observed evolution. However, interested readers can request the authors for the plots of all the cases analyzed. Returning to the objects released with the B1 initial conditions, Figs. 14–16 show the evolution of the mean semi-major axis over 100 years for several values of the area-to-mass ratio. The common feature of all the plots, which is shared by the other cases analyzed but not shown here, is that the semi-major axis remained close to the semisynchronous value, even when orbital decay was imminent (Figs. 15 and 18) due to the growth in eccentricity (Figs. 17–19). In addition, note that in this case (B1) an eccentricity growth capable of inducing an ‘‘earlier” reentry is possible even for intermediate values of the area-to-mass ratio (Figs. 13 and 18), around an ‘‘instability island” cencentered on A/M 25 m2/kg (Fig. 13). Finally, the long-term behavior of the mean inclination is shown in Figs. 20–22. For A/M 6 1 m2/kg the evolution of the orbit plane was still dominated by the interaction between J2 and the third body attraction, with the typical GPS Block II nodal regression period of 26 years and an inclination oscillation amplitude of 1.5°, superimposed on a longer term trend driven by luni-solar perturbations. An increase in A/M had, as a consequence, a faster nodal regression and wider amplitude of the inclination excursion. 5. Discussion To put all the results, including the examples presented in Section 4, into a wider context, the separate eﬀects of the various relevant perturbations were analyzed. For instance,

Fig. 13. Orbital lifetime, as a function of the area-to-mass ratio, for objects released, with negligible DV, in each of the six orbital planes used by the GPS constellation, according to the initial conditions described in Section 1.

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Fig. 14. Long-term evolution of the mean semi-major axis, for A/M = 0.05, 1, 5 and 10 m2/kg, assuming the B1 initial conditions. In each case the semimajor axis remained very close to the semi-synchronous one.

Figs. 23–25 show the impact of several combinations of perturbations on the evolution of the mean semi-major axis, eccentricity and inclination, respectively, assuming the B1 initial conditions and A/M = 25 m2/kg. From these

plots, the speciﬁc roles played by direct solar radiation pressure, geopotential harmonics and luni-solar attraction are clearly evident and the observed orbit evolution depends on their complex interplay.

Fig. 15. Long-term evolution of the mean semi-major axis, for A/M = 15, 20 and 25 m2/kg, assuming the B1 initial conditions. Even when an orbital decay occurred (for A/M = 25 m2/kg), due to the excessive growth of the eccentricity, the semi-major axis remained very close to the semi-synchronous one until reentry.

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Fig. 16. Long-term evolution of the mean semi-major axis, for A/M = 35 and 45 m2/kg, assuming the B1 initial conditions. Even for extremely large values of A/M (45 m2/kg) the semi-major axis remained very close to the semi-synchronous one for several decades, later gradually decreasing due to the periodic plunges of the perigee in the Earth’s atmosphere.

In order to show the relevance of the inclination lunisolar resonances, the same case was simulated by slightly

varying the initial B1 inclination i0 around the nominal value of 55.2°. The results are presented in Figs. 26–28

Fig. 17. Long-term evolution of the mean eccentricity, for A/M = 0.05, 1, 5 and 10 m2/kg, assuming the B1 initial conditions. Due to the interplay of solar radiation pressure, geopotential and luni-solar resonances, the trends observed, superimposed to the near yearly oscillations caused by radiation pressure, were quite complex and not easy to predict in advance.

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Fig. 18. Long-term evolution of the mean eccentricity, for A/M = 15, 20 and 25 m2/kg, assuming the B1 initial conditions. Due to the interplay of solar radiation pressure, geopotential and luni-solar resonances, the trends observed, superimposed on the near yearly oscillations caused by radiation pressure, were quite complex and not easy to predict in advance. Note the rapid growth of eccentricity, leading to an ‘‘earlier” reentry, induced by A/M = 25 m2/kg.

Fig. 19. Long-term evolution of the mean eccentricity, for A/M = 35 and 45 m2/kg, assuming the B1 initial conditions. Due to the interplay of solar radiation pressure, geopotential and luni-solar resonances, the trends observed, superimposed on the near yearly oscillations caused by radiation pressure, were quite complex and not easy to predict in advance. However, note the long-term stability of orbits with eccentricities up to about 0.7 (for A/ M = 45 m2/kg).

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Fig. 20. Long-term evolution of the mean inclination, for A/M = 0.05, 1, 5 and 10 m2/kg, assuming the B1 initial conditions. For A/M 6 1 m2/kg, the orbit plane evolution was still dominated by the interaction between the Earth’s oblateness and luni-solar perturbations, with the typical GPS nodal regression and inclination oscillation period of approximately 26 years. An increase in A/M resulted in a faster nodal regression and wider amplitude of the inclination oscillation, with small ripples of almost yearly frequency, caused by direct solar radiation pressure.

Fig. 21. Long-term evolution of the mean inclination, for A/M = 15, 20 and 25 m2/kg, assuming the B1 initial conditions. The increase in A/M resulted in a progressively faster nodal regression and wider amplitude of the inclination oscillation, with small ripples of almost yearly frequency, caused by direct solar radiation pressure.

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Fig. 22. Long-term evolution of the mean inclination, for A/M = 35 and 45 m2/kg, assuming the B1 initial conditions. The increase in A/M resulted in a progressively faster nodal regression and wider amplitude of the inclination oscillation, with small ripples of almost yearly frequency, caused by direct solar radiation pressure. Note that, for suﬃciently high A/M values, the orbit became periodically retrograde, i.e. with an inclination greater than 90°.

Fig. 23. Eﬀects of the perturbations on the mean semi-major axis evolution for an object with A/M = 25 m2/kg released with the B1 initial conditions. The perturbations considered were the direct solar radiation pressure with eclipses (SRP), the geopotential harmonics up to the 4th (4 4) and 16th (16 16) degree and order, and the lunar (Moon) and solar (Sun) third body attraction. The impact of luni-solar (mainly lunar) perturbations is evident.

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Fig. 24. Eﬀects of the perturbations on the mean eccentricity evolution for an object with A/M = 25 m2/kg released with the B1 initial conditions. The perturbations considered were the direct solar radiation pressure with eclipses (SRP), the geopotential harmonics up to the 4th (4 4) and 16th (16 16) degree and order, and the lunar (Moon) and solar (Sun) third body attraction. The role played by luni-solar (mainly lunar) resonances in increasing the eccentricity, leading to an ‘‘earlier” orbit decay, is evident.

Fig. 25. Eﬀects of the perturbations on the mean inclination evolution for an object with A/M = 25 m2/kg released with the B1 initial conditions. The perturbations considered were the direct solar radiation pressure with eclipses (SRP), the geopotential harmonics up to the 4th (4 4) and 16th (16 16) degree and order, and the lunar (Moon) and solar (Sun) third body attraction. In this case, the combined eﬀect of all the perturbations was to mitigate the inﬂuence of solar radiation pressure, which, if left alone, would lead to a main inclination oscillation of 47° (corresponding to ±23.5°, i.e. the value of the ecliptic obliquity) for any value of A/M.

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Fig. 26. Impact of the initial inclination on the mean semi-major axis evolution for an object with A/M = 25 m2/kg released with the B1 initial conditions. The importance of inclination luni-solar resonances in the GPS orbital regime is testiﬁed by the fact that a change of just 1° or 2° in the initial inclination was suﬃcient to avoid the ‘‘earlier” decay of the object.

Fig. 27. Impact of the initial inclination on the mean eccentricity evolution for an object with A/M = 25 m2/kg released with the B1 initial conditions. The importance of inclination luni-solar resonances in the GPS orbital regime is testiﬁed by the fact that a change of just 1° or 2° in the initial inclination was suﬃcient to limit eccentricity growth, preventing the ‘‘earlier” decay of the object.

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Fig. 28. Impact of the initial inclination on the mean inclination evolution for an object with A/M = 25 m2/kg released with the B1 initial conditions.

for the mean semi-major axis, eccentricity and inclination, respectively. They highlight that an increase of just 1° or, alternatively, a decrease of 2° are, in fact, suﬃcient to limit eccentricity growth (Fig. 27), preventing the ‘‘earlier” decay of the object (Fig. 26). To summarize the factors aﬀecting the orbit stability of high area-to-mass ratio objects released into GPS Block II trajectories, the orbits closer to the stable equilibrium points in the phase plane (kN, a) are more stable for area-to-mass ratios up to 45 m2/kg, while the trajectories may be chaotic close to the unstable equilibrium points. When solar radiation pressure acceleration is comparable to that of J2 (1– 2 104 m/s2, due to CR A/M = 20–30 m2/kg), luni-solar attraction induces a long-term growth in eccentricity which may lead, together with the other perturbations, to an ‘‘earlier” decay of orbits quite far from the stable equilibrium points (e.g. B1 and D1, as shown in Figs. 13 and 24). However, the details of the long-term evolution critically depend on the initial inclination, due to the inﬂuence of inclination luni-solar resonances. It should also be noted, as found by Ely (1999), that an intricate web of luni-solar resonances, depending on inclination and eccentricity, comes into play when the eccentricity and/or the inclination change substantially. This aﬀects the long-term evolution and, possibly, the chaotic behavior. The long-term evolution of high A/M objects released in GPS Block II orbits is thus very complex and highly sensitive to the initial conditions, due to geopotential and luni-solar resonances. Direct solar radiation pressure alone would basically induce a near yearly oscillation of eccentricity (e) and semi-major axis (a), whose amplitude would grow with

increasing A/M values: for CR A/M = 30 m2/kg, De 0.46 and Da ±48 km; for CR A/M = 42 m2/kg, De 0.50 and Da ±55 km. It would also lead to a long-term precession of the inclination vector, with a small yearly ‘‘nutation” superimposed, with a period ?1 for A/M ? 0 and ?0 for A/M ? 1 (30 years for CR A/M = 30 m2/kg, 10 years for CR A/M = 42 m2/kg). However, the amplitude of the inclination (i) oscillation would be 47° for any value of A/M, i.e. Di ±e, where e is the ecliptic obliquity 23.5°, as already found with geosynchronous objects (Valk et al., 2008; Valk, 2008). When acting together with the other perturbations, direct solar radiation pressure on high A/M objects would signiﬁcantly change both eccentricity and inclination, moving the state vector in the (i, e) plane through an intricate web of luni-solar resonances. This explains the complex evolution observed. In general, maximum eccentricity would increase with growing A/M values. For the E2 case, for example, maximum eccentricity would be 0.2 for CR A/M 10 m2/kg, 0.5 for CR A/M 30 m2/kg and 0.7 for CR A/M 50 m2/kg. However, certain initial conditions, coupled with luni-solar resonances, may change this simple pattern, adding to the yearly oscillation, due to the direct inﬂuence of solar radiation pressure, a signiﬁcantly longer period and wide amplitude eﬀect. This happened in the B1 and D1 cases, when an ‘‘earlier” decay due to e > 0.75 occurred with CR A/M 30 m2/kg and 40 m2/kg, respectively (Fig. 13). Therefore, depending on the initial conditions, a maximum ‘‘relatively stable” eccentricity of 0.7 would be obtained with 25 m2/ kg 6 CR A/M 6 90 m2/kg.

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Concerning the evolution of the orbit plane, as mentioned in Section 4, for A/M 6 1 m2/kg it would be still dominated by the interplay between the Earth’s oblateness and luni-solar perturbations, with the typical nodal regression period of approximately 26 years and an inclination oscillation amplitude of about 1.5°, superimposed on a longer term trend driven by third body attraction. An increase in A/M would result in a faster nodal regression and wider amplitude of the inclination oscillation, even though, for any given value of A/M, the nodal rate and the inclination excursion would depend on the initial conditions. In the E2 case, for example, CR A/M 12 m2/ kg would induce a Di ±3.5° and a nodal precession period of 20.6 years, CR A/M 30 m2/kg would induce a Di ±10° and a nodal precession period of 11.7 years, and CR A/M 48 m2/kg would induce a Di ±15° and a nodal precession period of 6.6 years. Note that, for sufﬁciently high A/M values and speciﬁc initial conditions, the orbit would become periodically retrograde (i.e. with i > 90°) for some time. For instance, this situation would occur with CR A/M P 48 m2/kg in the B1 case (Fig. 22) and with CR A/M P 54 m2/kg in the C4 case.

6. Conclusions This study has revealed and clariﬁed several qualitative and quantitative aspects concerning the long-term evolution of a new, and still hypothetical, class of orbital debris, characterized by very high area-to-mass ratios and released with negligible DV in the orbits used by GPS Block II satellites. Among many other results, mainly summarized in Section 5, it was found that also in this case, as with similar objects released in geostationary orbit, very long orbital lifetimes are possible, even with extremely high A/M values. Depending on the initial conditions, eccentricities as high as 0.7 could be attained with 25 m2/kg 6 CR A/M 6 90 m2/kg, maintaining, however, a semi-major axis and mean motion close to the semi-synchronous values. The inclination would also be subject to a wider excursion, linked to a faster regression of the ascending node, with increasing values of the area-to-mass ratio. In conclusion, if high A/M objects, such as those discovered in near synchronous orbits, have been released in GPS trajectories, they would have long enough lifetimes, in high eccentricity semi-synchronous orbits, to be detected by optical observers using appropriate search strategies.

Acknowledgments The results described in this paper were presented at the 37th COSPAR Scientiﬁc Assembly, held in Montre´al, Canada, on 13–20 July 2008. They were obtained within the framework of the ASI/CISAS Contract No. I/046/07/0.

References Agapov, V., Biryukov, V., Kiladze, R., Molotov, I., Rumyantsev, V., Sochilina, A., Titenko, V. Faint GEO objects search and orbital analysis, in: Danesy, D. (Ed.), Proceedings of the Fourth European Conference on Space Debris, ESA SP-587, ESA Publications Division, Noordwijk, The Netherlands, pp. 119–124, 2005. Anselmo, L., Pardini, C. Orbital evolution of geosynchronous objects with high area-to-mass ratios, in: Danesy, D. (Ed.), Proceedings of the Fourth European Conference on Space Debris, ESA SP-587, ESA Publications Division, Noordwijk, The Netherlands, pp. 279–284, 2005. Anselmo, L., Pardini, C. Long-Term Evolution of High Earth Orbits: Eﬀects of Direct Solar Radiation Pressure and Comparison of Trajectory Propagators, Technical Report 2007-TR-008, ISTI/CNR, Pisa, Italy, 29 March 2007. Beutler, G. Methods of Celestial Mechanics, Volume II: Application to Planetary System, Geodynamics and Satellite Geodesy. Springer, Berlin, Germany, 2005. Chao, C.C. Analytical investigation of GEO debris with high area-to-mass ratio, Paper No. AIAA-2006-6514, Presented at the 2006 AIAA/AAS Astrodynamics Specialist Conference, Keystone, Colorado, USA, 2006. Ely, T.A. Impact of eccentricity on East–West stationkeeping for the GPS class of orbits, Paper No. AAS-99-389, Presented at the 1999 AAS/ AIAA Astrodynamics Specialist Conference, Girdwood, Alaska, USA, 1999. Hugentobler, U.. Astrometry and Satellite Orbits: Theoretical Considerations and Typical Applications Geoda¨tisch-geophysikalische Arbeiten in der Schweiz, vol. 57. Schweizerische Geoda¨tische Kommission, Zu¨rich, Switzerland, 1998. Hughes, S. Earth satellite orbits with resonant lunisolar perturbations. I. Resonances dependent only on inclination. Proc. Royal Soc. Lond. A 372, 243–264, 1980. Ineichen, D., Beutler, G., Hugentobler, U. Sensitivity of GPS and GLONASS orbits with respect to resonant geopotential parameters. J. Geodesy 77, 478–486, 2003. Kwok, J.H. The Artiﬁcial Satellite Analysis Program (ASAP), Version 2.0, JPL NPO-17522, Jet Propulsion Laboratory (JPL), Pasadena, CA, USA, 20 April 1987. Liou, J.-C., Weaver, J.K. Orbital evolution of GEO debris with very high area-to-mass ratios. The Orbital Debris Quarterly News 8 (3), 6–7, 2004. Liou, J.-C., Weaver, J.K., Orbital dynamics of high area-to-mass ratio debris and their distribution in the geosynchronous region, in: Danesy, D. (Ed.), Proceedings of the Fourth European Conference on Space Debris, ESA SP-587, ESA Publications Division, Noordwijk, The Netherlands, pp. 285–290, 2005. Pardini, C., Anselmo, L. SATRAP: Satellite Reentry Analysis Program, Internal Report C94-17, CNUCE/CNR, Pisa, Italy, 30 August 1994. Pardini, C., Anselmo, L. Long-term evolution of geosynchronous orbital debris with high area-to-mass ratios. Trans. Jpn. Soc. Aero. Space Sci. 51, 22–27, 2008. Rossi, A. Resonant dynamics of medium Earth orbits: space debris issues. Celest. Mech. Dyn. Astr. 100, 267–286, 2008. Schildknecht, T., Musci, R., Ploner, M., Beutler, G., Flury, W., Kuusela, J., de Leon Cruz, J., de Fatima Dominguez Palmero, L. Optical observations of space debris in GEO and in highly-eccentric orbits. Adv. Space Res. 34, 901–911, 2004. Schildknecht, T., Musci, R., Flohrer, T. Properties of the high area-tomass ratio space debris population at high altitudes. Adv. Space Res. 41, 1039–1045, 2008. Valk, S. Global dynamics of geosynchronous space debris with high areato-mass ratios, Ph.D. Thesis in Mathematics, University of Namur, Belgium, 2008. Valk, S., Lemaıˆtre, A., Anselmo, L. Analytical and semi-analytical investigations of geosynchronous space debris with high area-to-mass ratios. Adv. Space Res. 41, 1077–1090, 2008.

Available online at www.sciencedirect.com

Advances in Space Research 43 (2009) 1509–1526 www.elsevier.com/locate/asr

Global dynamics of high area-to-mass ratios GEO space debris by means of the MEGNO indicator S. Valk, N. Delsate *, A. Lemaıˆtre, T. Carletti University of Namur (FUNDP), De´partement de Mathe´matique, Unite´ de Syste´mes Dynamiques, 8, Rempart de la Vierge, B-5000 Namur, Belgium Received 21 April 2008; received in revised form 6 February 2009; accepted 23 February 2009

Abstract In this paper we provide an extensive analysis of the global dynamics of high-area-to-mass ratios geosynchronous (GEO) space debris, applying a recent technique developed by Cincotta and Simo´ [Cincotta, P.M., Simo´, C.Simple tools to study global dynamics in nonaxisymmetric galactic potentials–I. Astron. Astrophys. (147), 205–228, 2000.], Mean Exponential Growth factor of Nearby Orbits (MEGNO), which provides an eﬃcient tool to investigate both regular and chaotic components of the phase space. We compute a stability atlas, for a large set of near-geosynchronous space debris, by numerically computing the MEGNO indicator, to provide an accurate understanding of the location of stable and unstable orbits as well as the timescale of their exponential divergence in case of chaotic motion. The results improve the analysis presented in Breiter et al. [Breiter, S., Wytrzyszczak, I., Melendo, B. Long-term predictability of orbits around the geosynchronous altitude. Advances in Space Research 35, 1313–1317, 2005] notably by considering the particular case of high-area-to-mass ratios space debris. The results indicate that chaotic orbits regions can be highly relevant, especially for very high area-to-mass ratios. We then provide some numerical investigations and an analytical theory that lead to a detailed understanding of the resonance structures appearing in the phase space. These analyses bring to the fore a relevant class of secondary resonances on both sides of the wellknown pendulum-like pattern of geostationary objects, leading to a complex dynamics. Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Solar radiation pressure; Space debris; MEGNO; Detection of chaos; Long-term evolution; Geosynchronous orbit; High area-to-mass ratios; Secondary resonances

1. Introduction Recent optical surveys in high-altitude orbits, performed by the European Space Agency 1 m telescope on Tenerife (Canary islands), have discovered a new unexpected population of 10 cm sized space debris in near geosynchronous orbits (GEO). These objects sometimes present highly eccentric orbits with eccentricities as high as 0.55 (Schildknecht et al., 2004; Schildknecht et al., 2005). Following the initial guess of Liou and Weaver (2004) who suggested that this new population may be constituted by GEO objects with high area-to*

Corresponding author. E-mail address: nicolas.delsate@ fundp.ac.be (N. Delsate).

mass ratios, recent numerical and analytical investigations were performed to support this assumption (Anselmo and Pardini, 2005; Liou and Weaver, 2005). In addition, these authors and others, such as Chao (2006) and later Valk et al. (2008), presented some detailed results concerning the short- and long-term evolution of high area-to-mass ratios geosynchronous space debris subjected to direct solar radiation pressure. More speciﬁcally, these latter authors mainly focused their attention on the long-term variation of both the eccentricity and the inclination vector. Moreover, some studies concerning the eﬀects of the Earth’s shadowing eﬀects on the motion of such space debris were given in Valk and Lemaıˆtre (2008). However, nobody ever dealt with the question to know whether these orbits

0273-1177/$36.00 Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2009.02.014

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are really predictable or not on the time scales of their investigations. The objective of this paper is twofold. The ﬁrst goal is the investigation of the long-term stability of high areato-mass ratio space debris subjected to direct solar radiation pressure, by means of the Mean Exponential Growth factor of Nearby Orbits (MEGNO) criterion. Second, still considering high area-to-mass ratios, we bring to the fore a relevant class of additional secondary structures appearing in the phase space. The paper is organized as follows. In Section 2, we focus our attention to the speciﬁcation of the underlying model and we give some details about the numerical aspects of the method. In Section 3, for the sake of completeness, we dwell upon the detailed deﬁnition of the MEGNO indicator, also providing a review of its main properties, in order to understand the behavior of the chaos indicator. Then in Section 4, in the framework of the validation of our implementation, we retrieve the results obtained by Breiter et al. (2005). We also discuss the signiﬁcance of the time of integration, recently reported by Barrio et al. (in press). In Section 5, we apply the MEGNO technique in order to give a insightful understanding of the stability of high area-tomass ratio space debris. More speciﬁcally, we show that the orbits of such peculiar space debris are extremely sensitive to initial conditions, especially with respect to the mean longitude and the semi-major axis. Second, we perform extended numerical analyses, showing that the related 2dimensional phase space is dominated by chaotic regions, in particular when the area-to-mass ratio is large. In addition, we also provide some results presenting the importance of the initial eccentricity value in the appearance of chaotic regions. Finally, in Section 6, we present extensive numerical and analytical investigations of the additional patterns which will be identiﬁed as secondary resonances. 2. The model For the purpose of our study, we consider the modeling of a space debris subjected to the inﬂuence of the Earth’s gravity ﬁeld, to both the gravitational perturbations of the Sun and the Moon as well as to the direct solar radiation pressure. As a consequence the diﬀerential system of equations governing the dynamics is given by €r ¼ apot þ

þ a þ arp ;

where apot is the acceleration induced by the Earth’s gravity ﬁeld, which can be expressed as the gradient of the following potential U ðr; k; /Þ ¼

1 X n n lX Re Pmn ðsin /ÞðC nm cos mk þ S nm sin mkÞ; r n¼0 m¼0 r

ð1Þ

where the quantities C nm and S nm are the spherical harmonics coeﬃcients of the geopotential. The Earth’s gravity ﬁeld adopted is the EGM96 model (Lemoine et al., 1987). In Eq.

(1), l is the gravitational constant of the Earth, Re is the Earth’s equatorial radius and the quantities ðr; k; /Þ are the geocentric spherical coordinates of the space debris. Pmn are the well-known Legendre functions. It is worth noting that the potential of Eq. (1) is subsequently expressed in Cartesian coordinates by means of the Cunningham algorithm (Cunningham, 1970). and a result from the gravity Both the accelerations interaction with a third body of mass m , where ¼ and ¼ , and can be expressed with respect to the Earth’s center of mass as a ¼ l

r r kr r k3

þ

r kr k3

! ;

where r and r are the geocentric coordinates of the space debris and of the mass m , respectively. The quantity l is the gravitational constant of the third-body. In our implementation, we chose the high accurate solar system ephemeris given by the Jet Propulsion Laboratory (JPL) to provide the positions of both the Sun and the Moon (Standish, 1998). Regarding direct solar radiation pressure, we assume an hypothetically spherical space debris. The albedo of the Earth is ignored and the Earth’s shadowing eﬀects are not taken into account either. The acceleration induced by direct solar radiation pressure is given by 2 a A r r ; arp ¼ C r P r kr r k m kr r k where C r is the adimensional reﬂectivity coeﬃcient (ﬁxed to 1 further on in this paper) which depends on the optical properties of the space debris surface; P r ¼ 4:56 106 N=m2 is the radiation pressure for an object located at the distance of 1 AU; a ¼ 1 AU is a constant parameter equal to the mean distance between the Sun and the Earth and r is the geocentric position of the Sun. Finally, the coeﬃcient A=m is the so-called area-to-mass ratio where A and m are the eﬀective cross-section and mass of the space debris, respectively. 3. The mean exponential growth factor of nearby orbits For the sake of clarity we present in this section the definition and some properties of the MEGNO criterion. Let Hðp; qÞ, with p 2 Rn ; q 2 Tn , be a n-degree of freedom Hamiltonian system and let us introduce the compact notation x ¼ ðp; qÞ 2 R2n as well as f ¼ ð@H=@q; @ H=@pÞ 2 R2n , then the dynamical system is described by the following set of ordinary diﬀerential equations d xðtÞ ¼ f ðxðtÞ; aÞ; dt

x 2 R2n ;

ð2Þ

where a is a vector of parameters entirely deﬁned by the model. Let /ðtÞ ¼ /ðt; x0 ; t0 Þ be a solution of the ﬂow deﬁned in Eq. (2) with initial conditions ðt0 ; x0 Þ, then it has

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associated the Lyapunov Characteristic Number (hereafter LCN), deﬁned by (Benettin et al., 1980) k ¼ lim

t!1

1 kd/ ðtÞk ; ln t kd/ ðt0 Þk

ð3Þ

where d/ ðtÞ, the so-called tangent vector, measures the evolution of an initial inﬁnitesimal deviation d/ ðt0 Þ d0 between /ðtÞ and a nearby orbit, and whose evolution is given by the variational equations (terms of order Oðd2 Þ are omitted) d d_ / ¼ d/ ðtÞ ¼ Jð/ðtÞÞ d/ ðtÞ; dt

with

Jð/ðtÞÞ ¼

@f ð/ðtÞÞ; @x ð4Þ

where Jð/ðtÞÞ is the Jacobian matrix of the diﬀerential system of equations, evaluated on the solution /ðtÞ. Let us note that the deﬁnition of LCN, given by Eq. (3), can also be written in an integral form Z 1 t d_ / ðsÞ ds; k ¼ lim t!1 t 0 d/ ðsÞ where d/ ¼ kd/ k; d_ / ¼ d_ / d/ =d/ . The Mean Exponential Growth factor of Nearby Orbits Y / ðtÞ is based on a modiﬁed time-weighted version of the integral form of LCN (Cincotta and Simo´, 2000). More precisely 2 Y / ðtÞ ¼ t

Z

t 0

d_ / ðsÞ s ds; d/ ðsÞ

as well as its corresponding mean value, to get rid of the quasi-periodic oscillation possibly existing in Y / ðtÞ Z 1 t Y / ðtÞ ¼ Y / ðsÞ ds: t 0 In the following we will omit the explicit dependence of Y and Y on the speciﬁc orbit /, when this will be clear from the context. Actually, Y ðtÞ allows to study the dynamics for long time scales, where generically Y ðtÞ does not converge, while limt!1 Y ðtÞ is well deﬁned (Cincotta et al., 2003). Consequently, the time evolution of Y ðtÞ allows to derive the possible divergence of the norm of the tangent vector dðtÞ, giving a clear indication of the character of the diﬀerent orbits. Indeed, for quasi-periodic (regular) orbits, Y ðtÞ oscillates around the value 2 with a linear growth of the separation between nearby orbits. On the other hand, for chaotic (irregular) motion, the norm of d grows exponentially with time, and Y ðtÞ oscillates around a linear divergence line. Cincotta et al. (2003) showed that, for the quasi-periodic orbits, Y ðtÞ always converges to 2, that is a ﬁxed constant. Moreover, it has been shown that ordered motions with harmonic oscillations, i.e. orbits very close to a stable periodic orbit, tend asymptotically to Y ðtÞ ¼ 0. These latter properties can also be used to compute eﬃciently a good estimation of LCN, or similarly the Lyapu-

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nov time T k ¼ 1=k, by means of a linear least square ﬁt of Y ðtÞ. Indeed, in the case of an irregular orbit, the time evolution of Y ðtÞ may be easily written as Y ðtÞ ’ aH t þ d;

t ! 1;

where aH is simply related to LCN by the relation aH ¼ k=2 and d is small for chaotic motion. But for regular orbits, the best ﬁtted linear regression, after a transient time, can provide a d not necessarily close to zero. Thus, the value of d may be considered as the measure of the time during which the orbits stick to a regular torus before getting chaotic (Cincotta and Simo´, 2000). Regarding the numerical computation of the MEGNO indicator, we adopt the same strategy as in Goz´dziewski et al. (2001). To be speciﬁc, in addition to the numerical integrations of both the equations of motion and the ﬁrst order variation equations, we consider the two additional diﬀerential equations d d_ d y¼ ; dt dd

d y w¼2 ; dt t

ð5Þ

which allow to derive the MEGNO indicators as Y ðtÞ ¼ 2 yðtÞ=t;

Y ðtÞ ¼ wðtÞ=t:

The MEGNO criterion, unlike the common Lyapunov variational methods, takes advantage of the whole dynamical information for the orbits and the evolution of its tangent vector, which results in shorter times of integration to achieve comparable results. Moreover, a couple of applications found in the literature (e.g. Goz´dziewski et al., 2001; Goz´dziewski et al., 2008; Cincotta and Simo´, 2000; Breiter et al., 2005) justify and conﬁrm that MEGNO is relevant, reliable and provides an eﬃcient way for the investigation of the dynamics by detecting regular as well as stochastic regimes. 3.1. MEGNO and numerical integrations As previously mentioned, in order to evaluate the MEGNO indicator, we have to integrate the diﬀerential system of Eq. (2), the linear ﬁrst order variational system of Eq. (4), as well as the two additional diﬀerential Eq. (5). We choose to write both the expressions of the perturbing forces and the variational system, i.e. the Jacobian matrix, in rectangular coordinates positions-velocities. In such a way we can overcome both the null eccentricity and the null inclination singularity present in the dynamics of space debris (Valk et al., 2009). Moreover, the explicit analytical expressions of the vector ﬁelds allow us to avoid the diﬃculties inherent in the classical method of neighboring trajectories (two particles method). In order to numerically integrate the two diﬀerential systems of equations, we adopt the variable step size Bulirsh–Stoer algorithm (see e.g. Bulirsh and Stoer, 1966; Stoer and Bulirsch, 1980). Let us note that, for the purpose

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of validation, the numerical integrations were also made with a couple of other numerical integrators. However, the Bulirsh–Stoer algorithm seems to be the best compromise between accuracy and eﬃciency. Moreover, as quoted by Wisdom (1983): What is more important for this study, Benettin et al. (1980) found that the maximum LCE1 did not depend on the precision of their calculation. It appears likely that as long as a certain minimum precision is kept, maximum LCE’s may be accurately computed, even though it is not possible to precisely follow a speciﬁed trajectory for the required length of time. Although this latter observation was formulated in the framework of both Lyapunov variational method and Hamiltonian systems, it seems that it remains relevant in the computation of the MEGNO criterion, at least in the particular case of our analysis.

eﬀects induced by the 1:1 resonance, the MEGNO evolution no longer depends on the random choice of the initial tangent vector. In this latter case, the intrinsic stability of the chosen orbits seems also to dictate the evolution of MEGNO as reported in Cincotta et al. (2003). More speciﬁcally, the stability of the orbit seems to inﬂuence the time evolution of MEGNO the more the orbit is closer to a stable or unstable equilibrium point. For instance, regarding the orbits extremely close to a stable equilibrium point, MEGNO generally approaches slowly the limit value 2 from below, even though some infrequent orbits present a MEGNO convergence from above. Conversely, the orbits initially close to the separatrices generally present a MEGNO approaching the value 2 from above.

3.2. Inﬂuence of the initial tangent vector d0

In this section we will study the MEGNO indicator for integrable Hamiltonian systems and we will show that generically (if the system is not isochronous) it always converges to 2, moreover the way Y ðtÞ reaches this limit value, say from higher or lower values, depends only on the choice of the initial tangent vector and not on the orbit itself. So let us consider an integrable Hamiltonian system write in action-angle variables, H ¼ HðpÞ, where p 2 B Rn denotes the action variables and q 2 Tn denotes the angle variables. Then the Hamiltonian equations are

By construction MEGNO depends on the initial value of the tangent vector d0 as the LCE (Benettin et al., 1980). This is why we preferred to adopt the strategy of randomly initialize the initial tangent vectors in order to avoid some parts of the artiﬁcially created zones of low MEGNO due to the proximity of d0 to the minimum Lyapunov exponent direction (Breiter et al., 2005). Moreover, as pointed out by Goz´dziewski et al. (2001), the random sampling of d0 is relevant in the sense that diﬀerent initial tangent vectors can lead to diﬀerent behaviors of the MEGNO time evolution while considering the same orbit. This observation has been reported in the framework of extra-solar planetary systems and seems to remain similar in the case of Earth orbiting objects and more generally for high-dimensional dynamical systems (having more than 3 degrees of freedom). Regarding the impact of the choice of the initial tangent vector d0 , we performed a set of exhaustive numerical investigations of regular orbits. More speciﬁcally, we compared the time-evolution of MEGNO using diﬀerent initial tangent vectors and identical generic initial conditions. The results conﬁrm that the random choice of the initial tangent vector induces a signiﬁcant random behavior in the way MEGNO approaches the limit value 2, hence preventing this information from being useful to check the stability/instability character of regular orbits. Actually, when considering a slightly perturbed two-body problem (such as the central attraction disturbed by the oblateness of the Earth), the way MEGNO converges to 2 is completely unpredictable, leading to more or less 50% of convergence of Y ðtÞ to 2 from above and the other remaining 50% from below. This result is formally discussed in the following subsections. However, when the order of magnitude of the perturbation is larger, the result does not completely hold anymore. In particular, when considering the perturbing

3.3. MEGNO for integrable systems

p_ ¼ 0; @H ¼ xðpÞ: q_ ¼ @p The tangent space (to a given orbit) can be split into the action and angle direction, namely d ¼ ðdp ; dq Þ, thus the variational system can be written as d_ p ¼ 0; 2

@ H dp ¼ MðpÞ dp : d_ q ¼ @p2 If the system is isochronous then M 0, thus dp and dq are constant and Y ðtÞ ¼ 0 for all t. On the other hand, if the system is non-isochronous we get dp ðtÞ ¼ dp ð0Þ and dq ðtÞ ¼ dq ð0Þ þ Mðpð0ÞÞ dp ð0Þt. To simplify the notations, let us introduce Mðpð0ÞÞ ¼ M 0 ;

Using the deﬁnition of MEGNO, we get

Y ðtÞ ¼ 1

Lyapunov Characteristic Exponent.

dp ð0Þ ¼ n0 and dq ð0Þ ¼ g0 :

1 t

Z

t 0

2

ðM 0 n0 Þ s þ M 0 n0 g0 ðn0 Þ2 þ ðg0 Þ2 þ 2M 0 n0 g0 s þ ðM 0 n0 Þ2 s2

s ds;

S. Valk et al. / Advances in Space Research 43 (2009) 1509–1526

@H @V ¼ @q @q @H @V ¼ xðpÞ þ ; q_ ¼ @p @p

and this integral can be explicitly computed, obtaining h i M 0 n0 g0 2 2 Y ðtÞ ¼ 2 log 1 þ 2M n g t þ ðM n Þ t 0 0 0 0 0 2 tðM 0 n0 Þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 ðM 0 n0 Þ ðM 0 n0 g0 Þ 2 t ðM 0 n0 Þ 2

p_ ¼

and a similar decomposition can be done for the variational system @2V @2V d p 2 dq d_ p ¼ @p@q @q 2 2 @ H @ V @2V d þ þ dq : d_ q ¼ p @p2 @p2 @p@q

2

M 0 n0 g0 þ ðM 0 n0 Þ t2 6 ﬃ 4arctan qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 ðM 0 n0 Þ ðM 0 n0 g0 Þ 3 M 0 n0 g0 7 ﬃ5: arctan qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 ðM 0 n0 Þ ðM 0 n0 g0 Þ

ð6Þ

One can check that the square root is well deﬁned, i.e. positive, and thus one can cast Eq. (6) into Y ðtÞ ¼ 2

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M 0 n0 g0 1 F 1 ðtÞ F 2 ðtÞ; t t

where F 1 and F 2 are positive functions and F 2 is bounded. We can then conclude that (see Fig. 1) (1) if M 0 n0 g0 > 0 then Y ðtÞ approaches 2 from below; (2) if M 0 n0 g0 < 0 then Y ðtÞ approaches 2 from above, in fact for large t the ﬁrst contribution dominates the bounded term F 2 . In this last part we will consider if and under which assumptions the previous results concerning the convergence Y ! 2 are still valid, for a quasi-integrable Hamiltonian system of the form H ðp; q; Þ ¼ H 0 ðpÞ þ V ðp; qÞ. The main idea is the following: ﬁx > 0, but small, and consider a ‘‘non-chaotic orbit” / , namely an orbit without a positive Lyapunov exponent (or with a bounded MEGNO), then if is suﬃciently small this orbit is a perturbation of an orbit existing also for ¼ 0; /0 , and we can check that Y / ¼ Y /0 þ OðÞ, hence the smallness of such -correction cannot change ‘‘the way Y goes to 2”. More precisely, the Hamilton equations are now

Looking for dp and dq as -power series, i.e. dp ¼ dp;0 þ dp;1 þ . . . and dq ¼ dq;0 þ dq;1 þ . . ., and collecting together, in the deﬁnition of MEGNO, terms contributing to the same power of , we can thus get 1 Y / ðtÞ ¼ t

Z

ðM 0 dp;0 Þ2 s þ M 0 dp;0 dq;0

t 0

2

ðdp;0 Þ þ ðdp;0 Þ2 þ 2M 0 dp;0 dq;0 s þ ðM 0 dq;0 Þ2 s2

s ds

þ OðÞ ¼ Y /0 ðtÞ þ OðÞ:

4. Validation of the method To validate our method we ﬁrst apply the technique on a simpliﬁed model, considering only the Earth’s gravity ﬁeld expanded up to the second degree and order harmonics, namely, J 2 ¼ C 20 ; C 22 and S 22 . For the purpose of the analysis, we followed a set of 12 600 orbits, propagated over a 30-year time span, that is the order of 104 fundamental periods (1 day) empirically required by the method (Goz´dziewski et al., 2001). As reported in Breiter et al. (2005), a 30-year time span seems to be relatively small for long-term investigations of geosynchronous space debris. However, the numerical integration of variational equations in addition to the extrapolation of the orbit is quite time consuming. Indeed, the simulation with an entry level step size of 400 s takes approximately 20 s per orbit when including only the Earth’s gravity ﬁeld, whereas it takes 42 s with a complete model. Thus, the examination of large sets of initial conditions can take a lot of time

Fig. 1. MEGNO for quasi-integrable adimensional Hamiltonian system. We consider the evolution of Y / for the system H ¼ p21 =2 þ p2 þ cos q1 þ cosðq1 q2 Þ. On the left panel ¼ 104 , while on the right panel ¼ 103 . In both cases is small enough to conﬁrm the theoretical predictions; let 1 0 observe that in this case the matrix M is given by and thus the sign condition reads Mdp;0 dq;0 ¼ d1p;0 d1q;0 . The unit of time corresponding to 1/10 0 0 of period of the orbit.

(typically 5 days for 104 orbits). On the other hand, the analysis of the following section will bring to the fore some indications about the Lyapunov times (smaller than 30 years). As a consequence, the integration time can be considered as suﬃciently large in the particular case of our study. For the purpose of this validation study, we consider a set of initial conditions deﬁned by a mean longitude k grid of 1°, spanning 90° on both sides of the ﬁrst stable equilibrium point and a semi-major axis a grid of 1 km, spanning the 42164 ± 35 km range. The other ﬁxed initial conditions are e0 ¼ 0:002 for the eccentricity, i0 ¼ 0:004 rad for the inclination, X0 ¼ x0 ¼ 0 rad for the longitude of the ascending node and the argument of perigee, respectively. These values have been ﬁxed to compare our results for the nearly-geosynchronous orbits with the ones of Breiter et al. (2005). As pointed out by Breiter et al. (2005), due to the 1:1 resonance, good variables to present our results will be ða0 ; r0 Þ, where a0 is the osculating initial semi-major axis and r is the so-called resonant angle, i.e. r ¼ k h, where h is the sidereal time. Fig. 2 (left panel) shows the MEGNO values computed using 30 years of integration time. We identify clearly a blow-up of the typical double pendulum-like pattern related to the 1:1 resonance. Here, we plot only over a horizontal range of 180°, i.e. only one eye. The existence of both the stable and the two unstable equilibrium points can be easily inferred. We observe that the phase space seems to be essentially ﬁlled in with MEGNO values Y ðtÞ ’ 2, that is plenty of regular orbits. Moreover, the two separatrices are also identiﬁable and are associated with neighboring MEGNO values 2 < Y ðtÞ 6 4. Therefore, following the properties deﬁned in Section 3, one could consider that these orbits are chaotic. However, we will show that this conclusion is false. Indeed, a careful identiﬁcation of the MEGNO time evolution shows that the latter always approach slowly the limit 2 from above. The closer to the separatrices, the slower the convergence. More precisely, orbits close to the separatrix integrated over long

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time span present a bounded MEGNO evolution. Hence they should be considered as non-chaotic. To clarify this point, we performed a similar study, but using a signiﬁcantly longer time-span, namely 300 years. The results are shown in Fig. 2 (right panel). For the sake of comparison, the color bars have been taken identical on both plots. Let us observe that the maximum value reached by the MEGNO is 4 in the left panel and 2.5 in the right one. In the 300 years simulation (Fig. 2, right), the MEGNO values, associated with orbits close to the separatrices, turn out to be, on average, smaller than in Fig. 2 (left panel), reaching almost the limit Y ðtÞ ! 2, due to the longer time of integration. Similarly, the dark zone in the neighborhood of the stable equilibrium point, corresponding to MEGNO values close to zero, is strongly shrunk, supporting the result that, in the limit of inﬁnitely large t, only the orbit originating from the exact stable equilibrium point leads to Y ¼ 0, whereas the neighboring trajectories converge slowly to Y ðtÞ ¼ 2. Let us note that the importance of the integration time has been recently reported by Barrio et al. (in press) in the framework of applications of the MEGNO method. We conﬁrm that a too short time of integration can give wrong conclusions about the dynamical behavior. Moreover, the latter paper also underlines some spurious structures appearing in the maps of the variational chaos indicators, explaining the presence of the sine wave of lower MEGNO with a bulge at the center of Fig. 2, ‘‘suggesting that the same periodic orbit is more or less regular depending on the initial conditions choice”. Actually, according to the latter authors and to our analysis, this conclusion is wrong because this spurious structure is related to numerical artifacts. 5. High area-to-mass ratios analysis The study of the long-term stability of near-geosynchronous objects has recently prompted an increasing interest

S. Valk et al. / Advances in Space Research 43 (2009) 1509–1526

of the scientiﬁc community. In the particular case of classical near-geosynchronous objects, the problem has been solved by computing the MEGNO indicator for a family of simulated geostationary, geosynchronous and super-geosynchronous orbits. A classical near-geosynchronous object has a period close to one sidereal day and is subjected to the main gravitational eﬀects of the Earth, including the 1:1 resonance, luni-solar perturbing eﬀects, as well as solar radiation pressure associated to a small area-tomass ratio ðA=m 1m2 =kgÞ. According to Breiter et al. (2005) and Wytrzyszczak et al. (2007), the near-geostationary region presents chaotic orbits only very close to the separatrices, due to the irregular transits between the libration and the circulation regimes. Regarding the super-geostationary orbits, all of them seem to be entirely regular on the time scale of the investigations, that is a few decades. The aim of this section is to provide a more extensive analysis of the dynamics of near-geosynchronous space debris with high area-to-mass ratios ðA=m 1m2 =kgÞ, subjected to direct solar radiation pressure. Our main objective is to study the eﬀects of high area-to-mass ratios on the stability of the principal periodic orbits and on the chaotic components. This analysis is divided into three parts. First, in Section 5.1, we focus our attention on the sensitivity to initial conditions; then, in Section 5.2, we report the results of dedicated numerical analyses which emphasize the importance of the area-to-mass ratio value. Finally, in Section 5.3, we study the inﬂuence of both the initial eccentricity and time at epoch. Let us recall that for large area-to-mass ratios ðA=m P 10m2 =kgÞ, the solar radiation pressure may become the major perturbation, by far larger than the dominant zonal gravity term J 2 (Valk et al., 2008). In this particular case, the larger the area-to-mass ratio, the more aﬀected the dynamics of the near-geosynchronous space debris, leading to daily high-amplitude oscillations of the semi-major axis, yearly oscillations of the eccentricity as well as long-term variations of the inclination. As an illustration, Fig. 3 shows the orbital elements histories of the ﬁrst 210 years of a geosynchronous high area-to-mass ratio space debris ðA=m ¼ 10 m2 =kgÞ. The yearly variation of the eccentricity reaches 0.2, which conﬁrms the expected values predicted (e.g. Anselmo and Pardini, 2005; Liou and Weaver, 2005). The inclination evolution presents a well known long-term variation whose period is directly related to the area-to-mass ratio value. Regarding the longitude of ascending node as well as the argument of perigee, they both present a libration due to the chosen set of initial conditions. For further details, we refer to Valk et al. (2008) as well as Chao (2006), where a full description of the longterm motion of high area-to-mass ratios space debris is given. 5.1. Sensitivity to initial conditions To start with, we follow the evolution of two high areato-mass ratio space debris ðA=m ¼ 10 m2 =kgÞ deﬁned by

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two sets of very close initial conditions, diﬀering only in the 10th digits in mean longitude. Fig. 3 shows the ﬁrst one and Fig. 4 shows the second nearby orbit. We observe that the most diﬀerence (in the behavior) take place in the semi-major axis and resonant angle panels. We notice that there are some diﬀerences in the dynamics of the semimajor axis already after 20 years. This is the same for resonant angle, conﬁrming the hypothesis that the sensitivity to initial conditions is especially relevant for the semimajor axis and resonant angle whereas the diﬀerence, in the behavior, between the other orbital elements remains small. We ﬁrst focus our attention on the time evolution of the semi-major axis and resonant angle. As a complement to Fig. 3, we numerically computed two orbits for two space debris with diﬀerent area-to-mass ratios, A=m ¼ 1 m2 =kg and A=m ¼ 10 m2 =kg, whose initial conditions have been chosen near the separatrices, to emphasize their chaotic behaviors. Fig. 5 shows a blow-up of the evolution of the semi-major axis (top panels) and resonant angle (middle panels) over the time span of 250 years. It is clear that the semi-major axis presents some irregular components over its evolution, related to some transitions between diﬀerent regimes of motion, clearly identiﬁable in the resonant angle plots. In addition, we also computed the corresponding MEGNO time evolution. The bottom panel in each graph shows the time evolution of the MEGNO indicator as well as its corresponding mean value. First, we see that the time evolution of Y ðtÞ presents a quasilinear growth almost since the beginning of the integration process, leading to the conclusion that these orbits are clearly chaotic over that time scale. Therefore, we also computed the linear ﬁt Y ðtÞ ’ aH t þ d in order to evaluate the Lyapunov time T k :T k is the inverse of the LCN ðkÞ calculated by the linear regression coeﬃcients aH ¼ k=2. Let us remark that to avoid the initial transient state, the least square ﬁts were performed on the last 85% of the time interval. This latter analysis brings to the fore the fact that larger area-to-mass ratios lead to smaller Lyapunov times, i.e. larger Lyapunov Characteristic Numbers. Indeed, for A=m ¼ 1m2 =kg, the Lyapunov time turns out to be on the order of 11 years, whereas it reaches the value T k ’ 3:7 years for A=m ¼ 10 m2 =kg. Second, let us also remark that the behavior of the MEGNO indicator is of particular interest in these cases. A careful analysis of Y ðtÞ underlines some irregular patterns directly related to the evolution of r, in particular when the orbits seem to transit across the separatrices. Finally, we can also highlight the fact that the sudden changes between libration and circulation regimes occur mainly when the inclination changes its sign of variation, especially at the maximum value for A=m 1 m2 =kg and at the minimum for A=m 6 1 m2 =kg (Fig. 5, top panels, dashed line), with an empirical long-term periodicity of T X , that is the long-term periodicity of the longitude of the ascending node, which is all the more smaller when A=m is large (Valk et al., 2008).

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Fig. 3. Time-evolution of high area-to-mass ratio space debris. Orbital elements over 210 years for a A=m ¼ 10 m2 =kg; initial conditions are: a0 ¼ 42166:473 km; e0 ¼ 0:002; i0 ¼ 0:004 rad, X0 ¼ x0 ¼ 0 rad and M 0 ¼ 4:928 rad. Time at epoch is 25 January 1991.

5.2. Extended numerical analyses We considered a set of 12,600 simulated orbits with various initial semi-major axes and mean longitudes. We took into account the following perturbing eﬀects: second degree and order harmonics (J 2 ; C 22 and S 22 ), the luni-solar interaction as well as the perturbing eﬀects of the solar radiation pressure with four values of the area-to-mass ratio ðA=m ¼ 1; 5; 10; 20 m2 =kgÞ. The results are reported in Fig. 6.

In the case with A=m ¼ 1 m2 =kg (top left panel) we recognize the same pendulum-like pattern as in Fig. 2. Considering the same integration time (30 years), we notice that the MEGNO values tend to be slightly larger than in Fig. 2 (left). Moreover, some irregularly distributed MEGNO values are clearly visible close to the two saddle unstable stationary points. These results completely agree with those presented by Breiter et al. (2005), where the solar radiation pressure was taken into account, but only for very small area-to-mass ratios (typically 0:005m2 =kg). Indeed, our lat-

Semi−Major Axis 4.222e+07

m

4.219e+07 4.216e+07 4.213e+07 4.210e+07 0

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Time [years] Eccentricity 0.255 0.185 0.115 0.045 −0.025 0

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ter analysis shows that in addition to the luni-solar perturbations, solar radiation pressure with small to moderate area-to-mass ratios, that is 0 6 A=m 6 1 m2 =kg, do not change considerably the phase space pattern. On the other hand, the remaining panels of Fig. 6 show that the phase portrait becomes signiﬁcantly more intricate with increasing area-to-mass ratios. Indeed, the width of the stochastic zone in the neighborhood of the separatrices becomes relevant, with a large displacement of the separatrices on the phase plane. The larger chaotic region can

readily be explained by the osculating motion of the separatrices due to the before-mentioned daily variations of the semi-major axis with respect to some mean value as well as by the increasing amplitudes of the eccentricities. These variations lead inevitably to transits through both the regions separating libration and circulation motion for orbits initially close to the separatrices. Moreover, it is also clear that the usual double pendulum-like phase space shows a tendency to be distorted with an apparent displacement of the unstable equilibrium

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points, whereas the stable equilibrium points remain almost ﬁxed. This last result is however quite awkward insofar as there is no physical interpretation to this phenomenon. Indeed, direct solar radiation pressure does not depend explicitly on the (mean) resonant angle with respect to the long-term investigations after averaging over short periodic terms. Therefore, it cannot induce a displacement of the equilibrium points in the phase space. Actually, a clever explanation can be found regarding the way the sampling is considered in the elaboration of the graphics. More speciﬁcally, it is worth noting that, at ﬁrst, the sampling is carried out with respect to osculating initial conditions.

Second, within the framework of mean motion theory, it is well-known that, due to the short-period oscillations, the mean and the osculating initial conditions cannot be considered to be equal. In other words, for the same ﬁxed value of the initial osculating semi-major axis and for various initial mean longitude, we obtain diﬀerent values for the mean semi-major axis; as explained with Fig. 7. Actually, the different initial mean longitudes induce a phase diﬀerence in the corresponding evolution of the semi-major axis, leading to diﬀerent mean initial semi-major axes. Let us remark that the maximum diﬀerence between both the mean semi-major axes is directly related to the order of magni-

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tude of the short-period variations, and, as a consequence, is also directly related to the area-to-mass ratio. More rigorously, the diﬀerence between osculating and mean initial conditions is a well-deﬁned transformation, depending on the generating function used within the averaging process allowing to change from mean to osculating dynamics. For further details concerning this explicit trans-

formation, we refer to the Lie algorithm discussed in Deprit (1969) and Henrard (1970). However, because we bound our analysis mainly to numerical simulations, we cannot access such generating function; we can nevertheless overcome this problem by numerically computing, for each semi-major axis osculating initial condition, the related mean initial semi-major axis, by considering the

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average over a short time span of 10 days. As an illustration, in Fig. 8, we give the relation between the mean semi-major axis and the resonant angle for various values of the osculating semi-major axis ðA=m ¼ 10 m2 =kgÞ. The ﬁrst diﬀerence is related to a semi-major axis sampling taken above the libration region, the second is related to a semi-major axis sampling which crosses the libration region and ﬁnally, the third sampling is taken below this region. In conclusion, we clearly see that the order of magnitude of the diﬀerences is, as previously mentioned, the order of the amplitudes of the daily variations observed in the semi-major axis dynamics. Let us note that in the latter case, i.e. A=m ¼ 10 m2 =kg, the diﬀerences reach at most 27 km, which correspond exactly to the diﬀerence between the stable and unstable equilibrium points, as shown in Fig. 6(bottom, left). We can thus apply numerically the transformation as a post-treatment process, that is considering the MEGNO val-

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5.3. Initial time at epoch and importance of the mean eccentricity One should also recall that solar radiation pressure leads to a theoretical equilibrium deﬁned both in eccentricity e0 and longitude of perigee -0 . The conditions leading to such an equilibrium can be written as ( e0 ¼ 32 C r P r mA n a1n cos2 2 ’ 0:01 C r mA ; -0 ¼ k ð0Þ:

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ues not in the osculating initial conditions phase space, but in the mean initial conditions phase space. For the sake of comparison with Fig. 6, we show the results once such a transformation has been applied (Fig. 9): it is clear that now the vertical gaps between both the stable and unstable equilibrium points are almost completely eliminated, hence these points have almost the same mean semi-major axis, getting rid of the what we called the ‘‘short-period artefact”. The thin light waves crossing the Fig. 9 are due to gaps in the set of initial conditions and have no dynamical signiﬁcance (also valid for the only one light wave crossing Fig. 13 in Section 6). Let us also remark that, from now on, all the results will be shown in the mean initial conditions phase space.

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Fig. 8. Relation between the mean semi-major axis and the resonant angle for various values of the osculating semi-major axis. The ﬁrst osculating semi-major axis is taken above the libration region, the second is related to an osculating semi-major axis sampling which crosses the libration region and, ﬁnally, the third sampling is taken below this region.

where n and n are the angular motions of both the space debris and the Sun, respectively, is the obliquity of the Earth with respect to the ecliptic and k ð0Þ the initial ecliptic longitude of the Sun. If these conditions are fulﬁlled, it has been shown (Chao, 2006; and later Valk et al., 2008, 2009), that the eccentricity vector ðe cos -; e sin -Þ remains constant, leading to a ﬁxed value of both the eccentricity and longitude of perigee. As an illustration, Fig. 10 shows the mid-term variations of the eccentricity for a ﬁxed value of the area-to-mass ratio ðA=m ¼ 10 m2 =kgÞ and ﬁxed initial conditions, namely, a0 ¼ 42; 164 km;e0 ¼ 0:1; i0 ¼ 0 rad, X0 ¼ x0 ¼ k0 ¼ 0 rad. It is clear that, apart from a phase diﬀerence, the amplitudes of the variations of the eccentricities are

Fig. 9. The MEGNO computed as a function of initial mean longitudes k0 and initial mean semi-major axis a0 . The model is the same as in Fig. 6. The areato-mass ratio is A=m ¼ 5 and 10 m2 =kg for the left and for the right graph, respectively.

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qualitatively the same, except when adopting an initial time at epoch equal to 21 March. In this latter case, the eccentricity remains almost constant, as expected by the theory. Fig. 11 shows the phase space in mean semi-major axis and longitude for A=m ¼ 10 m2 =kg and ﬁxed values of the initial conditions, namely e0 ¼ 0:1; i0 ¼ 0:004 rad, X0 ¼ x0 ¼ 0 rad. The diﬀerences between the two graphs only depends on the initial time at epoch parameter t0 . We could actually expect that diﬀerent initial times at epoch, namely, diﬀerent initial ecliptic longitudes of the Sun k ð0Þ, will reveal a quite rich collection of behaviors, depending on the diﬀerent states with respect to the before-mentioned eccentricity equilibrium. Actually, assuming an initial time at epoch of 21 December 2001, we see clearly that the phase space is ﬁlled by a large number of

chaotic orbits (Fig. 11, left). On the contrary, starting with an initial time at epoch of 21 March 2000, that is adopting a Sun pointing longitude of perigee (k ð0Þ ¼ 0 rad), the MEGNO values tend to be smaller and associated with significantly narrower chaotic regions, always located close to the separatrices (Fig. 11, right). In the latter case, the eccentricity presents only small yearly variations due to the proximity of the theoretical equilibrium. Therefore, these results seem to suggest that high amplitude variations of the eccentricity increase considerably the extension of chaotic regions close to the separatrices and, conversely, small eccentricity variations seem to minimize considerably the extent of chaotic regions. To justify this assumption, we performed a dedicated numerical simulation with the same set of parameters used in the one reported in Fig. 11, but considering higher values of the initial eccentricity. The results are reported in Fig. 12, the chosen time at epoch is 21 December 2000 and the initial eccentricities are, e0 ¼ 0:2 (left panel) and e0 ¼ 0:4 (right panel). In the latter case, the huge variations of the perigee altitude, induced by the large variations of the eccentricity as well as by the variations of the semi-major axis, leads to even more complicated dynamics. These results thus conﬁrm the importance of the initial eccentricity in the appearance of chaos. 6. Secondary resonances It is worth noting that inspecting Figs. 9, 11 and 12 we clearly note the presence of some additional patterns located on both sides of the separatrices in the phase space. These never seen before regions, hence unexplained so far, are actually characterized by very low MEGNO values. Indeed, this observation underlines the fact that the dynamics of high area-to-mass ratio space debris is even more intricate than expected. In the following two sections we will provide some numerical results and an analytical

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Fig. 12. MEGNO computed as a function of initial mean longitude k0 and semi-major axis a0 . The equations of motion include the central body attraction, the second degree and order harmonics J 2 ; C 22 and S 22 , the luni-solar interaction as well as the perturbing eﬀects of solar radiation pressure. The mean longitude grid is 1° and the semi-major axis grid is 500 m spanning the 42,164 ± 35 km range. The initial conditions are i0 ¼ 0:004 rad, X0 ¼ x0 ¼ 0 rad with A=m ¼ 10 m2 =kg. Time at epoch is 21 December 2000. The patterns have been obtained using two initial eccentricities, e0 ¼ 0:2 (left) and e0 ¼ 0:4 (right).

theory, based on a simpliﬁed model, to better understand such zones. 6.1. Numerical investigations We followed a large set of near-geosynchronous space debris, related to an extremely large set of initial conditions taken on both sides of the pendulum-like pattern, and for each one of the 72,000 orbits we computed the related MEGNO indicator. The initial conditions have been ﬁxed by a mean longitude grid of 1°, spanning 360°, and a semi-major axis grid of 1 km, spanning the 42,164 ± 100 km range, while the remaining orbit parameters and time at epoch are the same as in Fig. 6. Moreover, as in the previous extended analyses, the model of forces also includes the central body attraction, the second degree and order harmonics J 2 ; C 22 and S 22 as well as the combined attractions of the Sun and the Moon. The perturbing eﬀects of direct solar radiation pressure are also taken into account for a high area-to-mass ratio ﬁxed at 10 m2 =kg. The results are reported in Fig. 13, which is nothing but an extensive enlargement of the phase space presented in Fig. 6(bottom, left). This phase space widening clearly underlines the before-mentioned additional structures located at 40 km on each side of the resonant area. Furthermore, besides these patterns, what is of special interest is that this Figure also brings to the light supplementary structures located at approximately 80 km on both sides of the main resonance, suggesting that the phase space is actually foliated by a larger set of secondary structures. Moreover, the width of these additional patterns and the numerical values of the MEGNO both seem to be directly related to the inverse of the distance with respect to the resonant area. In addition, we also performed a set of similar numerical investigations, in order to distinguish qualitatively the relative relevance of some parameters such as the initial mean

Fig. 13. MEGNO computed as a function of initial mean longitude k0 and semi-major axis a0 . The equations of motion include the central body attraction, the second degree and order harmonics J 2 ; C 22 and S 22 as well as the luni-solar perturbations. The mean longitude grid is 1° and the semimajor axis grid is 1 km, spanning the 42,164 ± 100 km range. The initial conditions are e0 ¼ 0:002; i0 ¼ 0:004 rad and X0 ¼ x0 ¼ 0 rad. The areato-mass ratio is 10 m2 =kg. Time at epoch is 25 January 1991.

eccentricity, the value of the area-to-mass ratio, as well as the importance of the 1:1 resonance and of the third-body perturbations in the occurrence of such secondary structures. Even though these results are not presented here in detail, we can draw the following preliminary conclusions: the second order harmonic J 2 , as well as the third-body perturbations, do not seem to be really relevant and crucial in the appearance of these additional patterns. In other words, the unexpected patterns occur only when taking into account the combined eﬀects of both the second order and degree harmonic and direct solar radiation pressure. As a matter of fact, the extended numerical investigations performed in Fig. 6(top, left), or similarly those shown in

S. Valk et al. / Advances in Space Research 43 (2009) 1509–1526

Breiter et al. (2005), also present these structures, even though they are diﬃcult to perceive. Actually, the extension and chaoticity indicator of the secondary patterns seem to be directly proportional to the area-to-mass ratio value or, equivalently directly proportional to the mean value of the eccentricity. To get even more concluding results, we considered a blow-up of the phase space (dashed line rectangle in Fig. 13) with really high resolution sampling (150 m in the semi-major axis a and 0.3° in the resonant angle r). Fig. 14(top) shows this phase space widening wherein we deﬁned a so-called resonant angle section (horizontal black solid line), that is the subset of orbits having the same initial resonant angle value. This resonant angle section spans the complete range in semi-major axis and passes close to the stable equilibrium point. For each orbit deﬁned on this section, we computed the MEGNO indicator and in Fig. 14(middle) we report this value at the end of the simulation as a function of the semi-major axis. To double check our results, we performed a frequency analysis investigation (see Laskar, 1990; Laskar, 1995; Noyelles et al., 2008) aimed to study the behavior of the proper frequency of the resonant angle r, whose results are reported in Fig. 14(bottom). Here one can clearly notice the distinctive characteristics regarding the wellknown 1:1 resonance between the mean longitude and the

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sidereal time. Indeed, both MEGNO and the fundamental period show distinctively a minimum close to the stable equilibrium point. In this case, as previously mentioned in Section 4, MEGNO should slowly converge to Y ðtÞ ¼ 2 everywhere, except at the equilibrium point where the limit value is Y ðtÞ ¼ 0, that is why, using a ﬁnite integration time, we obtain such V-shaped curve, close to 0 in the center of the resonance and to 2 on the borders. It is also worth noting that the fundamental period of r is reported to be close to 2.25 years, which is in good agreement with the well-known 818 days libration period of a typical uncontrolled near-geosynchronous object. Near the separatrices, MEGNO clearly presents some obvious high values which conﬁrms the presence of chaotic orbits. Here, the fundamental period reaches signiﬁcant values and, as a matter of fact, is not well determined, once again supporting the result of the existence of a chaotic zone. Moreover, the use of frequency analysis allows us to support strongly the hypothesis that the additional patterns are actually related to secondary resonances. Indeed, if we look at the evolution of the fundamental period with respect to the semi-major axis, it is clear that the so-called secondary resonances are associated, regarding the angle r, with periods which are commensurate with 1 year. More precisely, the major secondary resonances, located at approximately 40 km on both sides of the pendulum-like

Fig. 14. Blow-up of the phase space with the speciﬁcation of a resonant angle section (horizontal black solid line), that is the set of orbits having the same ¼ 81:67 (top panel). Evolution of MEGNO with respect to the (osculating) initial resonant angle value, near the ﬁrst stable equilibrium point, namely rsection 0 initial semi-major axis a0 for the speciﬁed section (middle panel). The fundamental period of r with respect to the initial semi-major axis a0 , computed by means of frequency analysis for the speciﬁed section (bottom panel). The estimation of the periods are made over a 20 years period of time.

2σ + λ

sun

Resonant angle σ

pattern, are related to a 2-year fundamental period of r. Concerning the farther patterns located at 80 km, the fundamental period of r turns out to be very close to 1 year. As a consequence, we can presumably assume that these secondary resonances are actually related to a commensurability between r and the 1 year period angle k , that is the ecliptic longitude of the Sun. To justify this assumption, we focused our attention to the major secondary resonances located at 40 km on both sides of the pendulum-like pattern, considering the time evolution of various linear combinations of r and k . For this purpose, we considered various initial semi-major axes in the phase space. The results are shown in Fig. 15. At ﬁrst glance, it is apparent that three propagations stand apart from the others. In the ﬁrst row of Fig. 15, that is regarding the evolution of the resonant angle r, we clearly identify the well-known characteristics related to the primary resonance. In particular, in Fig. 15a, that is when considering an initial semi-major axis inside the primary resonant ða0 ¼ 42; 188 kmÞ; r shows the well-known longperiod libration (2.25 years), whereas r circulates outside this region. Furthermore, what is of special interest is the time evolution of both 2r þ k and 2r k , shown in the second and third row, respectively. It is clear that most of the time these angles show a circulation regime. However, when considering an initial semi-major axis inside the major lower secondary resonance for 2r k or, similarly inside the major upper secondary resonance for 2r þ k , both these angles show a signiﬁcant long-term evolution (Fig. 15b and c).

The presence and the location of these secondary resonances can be studied using an appropriate simpliﬁed model. Hence we model the averaged geostationary motion by a pendulum-like system, given by its Hamiltonian formulation (Valk et al., 2009) up to order e2 in the series expansion l2 l4 2 5 2 _ H ¼ 2 hL þ 3 6 Re 1 e S 2200 ðX; x; M; hÞ; 2 2L L where pﬃﬃﬃﬃﬃﬃ L ¼ la

and S 2200 ðX; x; M; hÞ ¼ C 22 cos 2r þ S 22 sin 2r:

In the context of direct solar radiation pressure, we can introduce the factor Z proportional to A=m through the eccentricity e (for further details, we refer to the averaged simpliﬁed analytical model developed in Valk et al., 2008, 2009). As a ﬁrst approximation, the time evolution of both the eccentricity e and the longitude of perigee - were found to be (neglecting the obliquity of the Earth with respect to the ecliptic) Z cos k þ a0 ; L n Z e sin - ¼ sin k b0 ; L n e cos - ¼

which introduces k in the Hamiltonian. The quantity n is the mean motion of the Sun and both a0 and b0 are related

6

6

6

6

4

4

4

4

2

2

2

2

0

0

0

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0

6

6

6

4

4

4

2

2

2

0

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2σ − λsun

6.2. Analytical investigation – simpliﬁed model

10

20

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6

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2

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to the initial conditions with respect to the eccentricity and the longitude of perigee). The resulting Hamiltonian takes the generic form l2 _ þ F cos ð2r 2r0 Þ G 2 cos ð2r hL 2 L6 L6 2L 2r0 Þ cos ðk þ dÞ;

H¼

where d; F ; G; r0 are constants. A suitable transformation is then necessary to introduce action-angle variables (w; J Þ in the libration and in the circulation region of the double pendulum, in such a way any trajectory of the double pendulum is characterized by a constant action J and a corresponding _ Rewriting the perturbed system (beconstant frequency w. cause of k terms) by means of these new variables and then using the expansions in Bessel functions, we could isolate any resonance of the type kw k in the circulation region, for any jkj, and in the libration region, for jkj P 3, which corresponds to our frequency analysis. This analysis is surely promising, but it is outside the goals of this paper. Further investigations will be detailed in a forthcoming publication (Lemaıˆtre et al., accepted for publication). 7. Conclusions The predictability of the trajectory high area-to-mass ratio space debris located near the geosynchronous region was investigated by means of a recent variational chaos indicator called MEGNO. Thanks to this technique, we clearly identiﬁed the regular (stable) and irregular (chaotic) orbits. This eﬃcient method allowed us to obtain a clear picture of the phase space, hence showing that chaotic regions can be particularly relevant, especially for very high area-to-mass ratio objects. Moreover, we discussed the importance of both the initial eccentricity and time at epoch in the appearance of chaos. Finally, we brought to the fore a relevant class of additional unexpected patterns which were identiﬁed as secondary resonances, that were numerically studied by means of both the MEGNO criterion and frequency map analysis, to eventually conclude that they involve commensurabilities between the primary resonant angle and the ecliptic longitude of the Sun. We also presented an analytical scheme that could explain their existence. It will be the subject of further work. Acknowledgements The authors thank S. Breiter for helpful discussions about both the MEGNO criterion as well as numerical issues which led to substantial improvement of the present paper, also providing some useful references. We are also grateful for the opportunity to use the frequency analysis tools developed by B. Noyelles and A. Vienne. Finally, the authors warmly thank the two referees for their suggestions that allowed us to strongly improve the paper.

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References Anselmo, L., Pardini, C. Orbital evolution of geosynchronous objects with high area-to-mass ratios, in: Danesy, D. (Ed.), Proceedings of the Fourth European Conference on Space Debris, (ESA SP-587). ESA Publications Division, Noordwijk, The Netherlands, pp. 279–284, 2005. Barrio, R., Borczyk, W., Breiter, S. Spurious structures in chaos indicators maps. Chaos Soliton. Fract., doi:10.1016/j.chaos.2007.09.084, in press. Benettin, G., Galgani, L., Giorgilli, A., et al. Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: a method for computing all of them. Part 1: Theory. Meccanica 15, 9–20, 1980. Breiter, S., Wytrzyszczak, I., Melendo, B. Long-term predictability of orbits around the geosynchronous altitude. Adv. Space Res. 35, 1313– 1317, 2005. Bulirsh, R., Stoer, J. Numerical treatment of ordinary diﬀerential equations by extrapolation methods. Numer. Math. 8, 1–13, March, 1966. Chao, C.C. Analytical investigation of GEO debris with high area-to-mass ratio. AIAA Paper No. AIAA-2006-6514, Presented at the 2006 AIAA/AAS Astrodynamics Specialist Conference, Keystone, Colorado, August 2006. Cincotta, P.M., Giordano, C.M., Simo´, C. Phase space structure of multidimensional systems by means of the mean exponential growth factor of nearby orbits. Physica D 182, 151–178, 2003. Cincotta, P.M., Simo´, C. Simple tools to study global dynamics in nonaxisymmetric galactic potentials – I. Astron. Astrophys. (147), 205– 228, 2000. Cunningham, L.E. On the computation of the spherical harmonics terms needed during the numerical integration of the orbital motion of an artiﬁcial satellite. Celest. Mech. 2, 207–216, 1970. Deprit, A. Canonical transformations depending on a small parameter. Celest. Mech. 1, 12–30, 1969. Goz´dziewski, K., Bois, E., Maciejewski, A.J., et al. Global dynamics of planetary systems with the MEGNO criterion. Astron. Astrophys. 378, 569–586, 2001. Goz´dziewski, K., Breiter, S., Borczyk, W. The long-term stability of extrasolar system HD37154, numerical study of resonance eﬀects. Mon. Notices RAS 383, 989–999, 2008. Henrard, J. On a perturbation theory using lie transforms. Celest. Mech. 3, 107–120, 1970. Laskar, J. The chaotic motion of the solar system: a numerical estimate of the size of the chaotic zones. Icarus 88, 266–291, 1990. Laskar, J. Introduction to frequency map analysis, in: Proceedings of 3DHAM95 NATO Advanced Institute, vol. 533, S’Agaro, 134–150, June 1995. Lemaıˆtre, A., Delsate, N., Valk, S. A web of secondary resonances for large A=m geostationary debris. Celestial Mechanics and Dynamical Astronomy, accepted for publication. Lemoine, F.G., Kenyon, S.C., Factor, J.K., et al. The development of the joint nasa gsfc and nima geopotential model EGM96. Tech. Rep., NASA, TP-1998-206861, 1987. Liou, J.-C., Weaver, J.K. Orbital evolution of GEO debris with very high area-to-mass ratios. The Orbital Quarterly News, vol. 8(3), The NASA Orbital Debris Program Oﬃce, 2004. Liou, J.-C., Weaver, J.K. Orbital dynamics of high area-to-mass ratio debris and their distribution in the geosynchronous region, in: Danesy, D. (Ed.), Proceedings of the Fourth European Conference on Space Debris (ESA SP-589). ESA Publications Division, Noordwijk, The Netherlands, pp. 285–290, 2005. Noyelles, B., Lemaıˆtre, A., Vienne, A. Titan’s rotation. A 3-dimensional theory. Astron. Astrophys. 475, 959–970, 2008. Schildknecht, T., Musci, R., Flury, W., et al. Optical observations of space debris in high-altitude orbits, in: Danesy, D. (Ed.), Proceedings of the Fourth European Conference on Space Debris. ESA SP-587. ESA Publications Division, Noordwijk, The Netherlands, pp. 113–118, 2005.

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Schildknecht, T., Musci, R., Ploner, M., et al. Optical observations of space debris in GEO and in highly-eccentric orbits. Adv. Space Res. 34, 901–911, 2004. Standish, E.M. JPL planetary and lunar ephemeris, DE405/LE405. JPL Interoﬃce Memorandum IOM 312.D-98-048, August 1998. Stoer, J., Bulirsch, R. Introduction to Numerical Analysis. SpringerVerlag, New York, 1980. Valk, S., Lemaıˆtre, A., Anselmo, L. Analytical and semi-analytical investigations of geosynchronous space debris with high area-to-mass ratios inﬂuenced by solar radiation pressure. Adv. Space Res. 41, 1077–1090, 2008.

Valk, S., Lemaıˆtre, A. Semi-analytical investigations of high area-to-mass ratio geosynchronous space debris including earth’s shadowing eﬀects. Adv. Space Res. 42 (8), 1429–1443, 2008. Valk, S., Lemaıˆtre, A., Deleﬂie, F. Semi-analytical theory of mean orbital motion for geosynchronous space debris under gravitational inﬂuence. Adv. Space Res. 43, 1070–1082, 2009. Wisdom, J. Chaotic behavior and the origin of the 3/1 Kirkwood gap. Icarus 56, 51–74, 1983. Wytrzyszczak, I., Breiter, S., Borczyk, W. Regular and chaotic motion of high altitude satellites. Adv. Space Res. 40, 134–142, 2007.

Available online at www.sciencedirect.com

Advances in Space Research 43 (2009) 1527–1531 www.elsevier.com/locate/asr

Faster algorithm of debris cloud orbital character from spacecraft collision breakup Li Yi-yong *, Shen Huai-rong, Li Zhi Academy of Equipment Command & Technology, Huairou City, Beijing, China Received 18 June 2008; received in revised form 3 March 2009; accepted 6 March 2009

Abstract Space debris is polluting the space environment. Collision fragment is its important source. NASA standard breakup model, including size distributions, area-to-mass distributions, and delta velocity distributions, is a statistic experimental model used widely. The general algorithm based on the model is introduced. But this algorithm is diﬃcult when debris quantity is more than hundreds or thousands. So a new faster algorithm for calculating debris cloud orbital lifetime and character from spacecraft collision breakup is presented ﬁrst. For validating the faster algorithm, USA 193 satellite breakup event is simulated and compared with general algorithm. Contrast result indicates that calculation speed and eﬃciency of faster algorithm is very good. When debris size is in 0.01–0.05 m, the faster algorithm is almost a hundred times faster than general algorithm. And at the same time, its calculation precision is held well. The diﬀerence between corresponding orbital debris ratios from two algorithms is less than 1% generally. Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Space debris; Orbital character; Collision breakup; USA 193; EVOLVE software

1. Introduction Research on space debris has been intensiﬁed in recent years as the number of space debris objects grows. The number of all objects in Earth orbit oﬃcially cataloged by the U.S. Space Surveillance Network (SSN) is more than 12,000 at the end of 2007 (The NASA Orbital Debris Program Oﬃce, 2008). The population of debris has now reached the level that orbital debris has become an important design factor for spacecraft. They induce serious space environment question and make potential threat to orbital spacecraft by the large impact energy. Although the current near-Earth space debris environment is dominated by explosion fragments, it is predicted that more debris will be generated by collisions rather than explosions in the future (Liou, 2006). Therefore, a fundamental issue in trying to limit the growth of future debris populations is to understand the nature of the predicted *

Corresponding author. Tel.: +86 010 66364384. E-mail address: [email protected] (L. Yi-yong).

orbital collisions. A key question is to have a high ﬁdelity breakup model and a faster and eﬀective algorithm to simulate collision activities. Since the 1970s, the NASA Orbital Debris Program Oﬃce has modeled the debris clouds generated by on-orbit explosions and collisions in terms of fragment size and velocity distributions. Up to now, the last major changes to these breakup models occurred in the 1990s. In the past ten years, NASA standard breakup model has been used to describe the outcome of spacecraft fragments widely (Michael et al., 2006). This paper presents a new algorithm of debris cloud orbital character from spacecraft collision breakup based on NASA standard breakup model (Reynolds et al., 1998). The USA 193 test is calculated based on the new algorithm and general algorithm separately, and the result validates that the new algorithm is faster and eﬀective. 2. Collision breakup model and algorithm A collision breakup model, at a minimum, should deﬁne the size, area-to-mass ratio, and ejection velocity of each

0273-1177/$36.00 Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2009.03.008

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generated fragment. Since these parameters are not constant for all debris, distributions as a function of a given parameter, e.g., mass or characteristic length, are necessary. In addition, the initial conditions of the breakup, e.g., the total mass of the parent objects or the collision velocity, can be highly inﬂuential. Since multiple breakup events of the same type of object, for instance a Delta second stage, will not produce exactly the same debris cloud each time, the breakup model should also address variances about the derived distributions. NASA standard breakup model was adopted in recent versions of EVOLVE (Johnson et al., 2001) and LEGEND (Liou et al., 2004). The new model was founded on broader experimental databases. The single-valued functions for number, area-to-mass ratio, and velocity were to be replaced by more representative distributions. And the new NASA approach employed ‘characteristic length’, Lc, as independent variable replacing previous mass. This leads to a fundamentally diﬀerent data ﬂow (Bendisch et al., 2004). But the new model has also drawbacks, for example, the masses assigned to objects below 1 mm diameter are too large, leading to very dense particles, and the additional velocity distribution for small fragments which does not match measurement data. Michael et al. (2004) found these shortcomings and presented the approach for the improvement of the breakup model fortunately, then applied it for MASTER 2005. Based upon NASA standard breakup model, the spacecraft collision breakup algorithm can be given (see Fig. 1) (Li et al., 2008). For convenient contrast, we can name this algorithm for general algorithm. The steps are as follows:

(1) To calculate size of every debris. Firstly, the type of collision (catastrophic or non-catastrophic) is decided by the target mass mt, projectile mass mp, and collision velocity v. Size distribution is calculated based on size distribution function. (2) To calculate A/M of every debris. Area-to-mass distribution is made certain for a given characteristic length based on area-to-mass ratio distribution function. (3) To calculate mass of every debris. Mass of a given size and A/M debris is calculated. A/M of every debris is independent and submits to certain distribution, so random calculated mass of debris is often uncertain and mass sum of all created debris likely does not meet mass of parent objects. Thus, recalculation to return (2) is required. (4) To calculate DV of every debris. Delta velocity distribution is got for a given A/M based on delta velocity distribution function. The validation of conservation of momentum and kinetic energy is needed. (5) To calculate orbit of every debris. Position, velocity and orbit parameters of every debris are obtained based upon DV of debris, breakup position and velocity. These parameters can be applied for calculating debris orbit motion. It’s obvious that every debris needs to be calculated for predicting its orbital character according to this algorithm. When the number of debris is not so large, it is convenient and eﬀective. However, the number of debris from spacecraft collision breakup is very large. Generally speaking, the number of debris over 10 cm size is thousands, and that debris of centimeters is tens thousand, and smaller debris is more. So this algorithm is diﬃcult because of huge data quantity and long calculation time, and it is necessary to ﬁnd a faster and eﬀective algorithm. 3. Faster algorithm of debris cloud orbital character NASA standard breakup model presents size distribution function of debris from spacecraft collision breakup. A power law relation exists between the number and size of debris. As numerical calculation, the size of debris can be divided into multi-section, Lci [Lc(i), Lc(i + 1)) (i = 1, 2, . . .). So the number of debris in every size section, N(Lci), can be calculated. N ðLci Þ ¼ N ðLc ðiÞÞ N ðLc ði þ 1ÞÞ ¼ 0:1ðMÞ0:75 Lc ðiÞ1:71 Lc ði þ 1Þ1:71

Fig. 1. The spacecraft collision breakup algorithm.

ð1Þ

where the characteristic length Lc = (x + y + z)/3 (in m); x is the longest linear body dimension, y is the longest body dimension orthogonal to x, and z completes the triad and is measured normal to the x–y plane. The value of M is deﬁned as the mass (in kg) of both objects in a catastrophic collision. In the case of a non-catastrophic collision, the

L. Yi-yong et al. / Advances in Space Research 43 (2009) 1527–1531

value of M is deﬁned as the mass (in kg) of the projectile multiplied by the collision velocity (in km/s). Then, area-to-mass ratio of debris is divided into multisection, A/Mj [A/M(j), A/M(j + 1)) (j = 1, 2, . . .). According to area-to-mass ratio distribution function, distribution probability of debris with A/Mj in the section of Lci, PA=M (Lci, A/Mj), can be calculated.

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Finally, orbital lifetime of representative debris in LEO, Lt (A/Mj, DVk, dirr), can be calculated by taking into account air drag (Vallado, 2004), and orbital character of debris can be analyzed. At the time of post-breakup t, in the size range of [Lc(i), Lc(i + 1)), the ratio of orbital debris to breakup debris is: X P ðt; Lci Þ ¼ P A=M ðLci ; A=M j ÞP DV ðA=M j ; DV k ÞP dir ðDV k ; dirr Þ

P A=M ðLci ; A=M j Þ ¼ F A=M ðkci ; vj Þ

j;k;r

ð5Þ

S=C

DA=M ðkci ; vj Þ jvjþ1 vj j ¼P S=C j DA=M ðkci ; vj Þ jvjþ1 vj j

ð2Þ

where DS=C A=M ðkc ; vÞ is the area-to-mass distribution function for spacecraft collision fragments based on NASA’s standard breakup model, kc ¼ log10 ðLc Þ, and v = log10(A/ M) is the variable in the area-to-mass distribution. Similarly, delta velocity of debris can be divided into multi-section, DVk [DV(k), DV(k + 1)) (k = 1, 2, . . .). According to delta velocity distribution function, distribution probability of debris with DVk in the section of A/Mj, PDV(A/Mj, DVk), can be calculated. P DV ðA=M j ; DV k Þ ¼ F DV ðvj ; mk Þ DCOLL ðvj ; mk Þðmkþ1 mk Þ ¼ P DVCOLL ðvj ; mk Þðmkþ1 mk Þ k DDV

ð3Þ

where, DCOLL ðv; mÞ is the delta velocity distribution funcDV tion for spacecraft collision fragments based on NASA’s standard breakup model, and m = log10(DV) is the variable in the delta velocity distribution. In the same way, an equal-angle subdivided polyhedron grid, such as Icosahedron, can be adopted for debris delta vp is velocity velocity directions (see Fig. 2). In Fig. 2, ~ vt ;~ vector of target and projectile, respectively, and ~ vcm is the equivalent velocity vector of system in the case of conservation of momentum. According to delta velocity direction equal-angle distribution, distribution probability of debris with dirr (r = 1, 2, . . .) in the section of DVk, Pdir(DVk, dirr), can be calculated. P dir ðDV k ; dirr Þ ¼

1 maxðrÞ

ð4Þ

where j, k and r should meet the case, Lt (A/Mj, DVk, dirr) > t. Obviously, in this size range, the number of orbital debris is N ðt; Lci Þ ¼ N ðLci ÞP ðt; Lci Þ

ð6Þ

4. A case calculation and analysis USA 193 was an American military satellite launched on December 14, 2006, and its precise function and purpose were classiﬁed. The satellite malfunctioned shortly after deployment. On February 14, 2008, U.S. oﬃcials announced the plan to destroy USA 193 before atmospheric reentry, stating that the intention was ‘‘saving or reducing injury to human life”. On February 21, 03:29 GMT U.S. navy employed a SM-3 missile intercepting the USA 193 satellite successfully. The satellite weight is 2270 kg, circle orbit, and height 247 km. The interceptor weight is about 30 kg, ﬂight velocity 2.6667 km/s. Their collision velocity reached 9.8333 km/s (Wikipedia, 2008). According to size distribution function, the number of debris in various size sections is calculated (see Table 1). It is shown that the number of debris rises rapidly as size decreases. As calculating debris cloud by the faster algorithm, we subdivide area-to-mass ratio into 103–102.5, 102.5– 102, 102–101.5, 101.5–101, 101–100.5, 100.5–100, 100–100.5, and 100.5–101, eight sections (include almost all debris area-to-mass ratio). At the same time, delta velocity is subdivided into 0–100.5, 100.5–100, 100–100.5, 100.5–101, 101–101.5, 101.5–102, 102–102.5, 102.5–103, 103–103.5, and 103.5–104, 10 sections (include almost all debris delta velocity). An equal-angle subdivided Icosahedron grid will be adopted for debris delta velocity directions. Supposing that survival height of debris is 100 km, atmosphere density refers to USSA76 model, drag coeﬃcient CD = 2.2. At the time of post-breakup t, in various size ranges, the ratio of orbital debris to breakup debris and the number of orbital Table 1 The number of debris in various size sections from USA 193 test.

Fig. 2. Sketch map of debris delta velocity direction.

Debris size (m)

Debris number (piece)

Accumulative number (piece)

P1.0 0.5–1.0 0.1–0.5 0.05–0.1 0.01–0.05

33 76 1594 3870 81,784

33 109 1703 5573 87,357

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Table 2 Space debris ratio (unit: %) and number (unit: piece) from faster algorithm. Post-breakup time

Debris size (m) 0.01–0.05

0h 2h 1 Day 7 Days 30 Days 60 Days 180 Days

0.05–0.1

0.1–0.5

0.5–1.0

P1.0

Ratio

Number

Ratio

Number

Ratio

Number

Ratio

Number

Ratio

Number

100 16.30 9.18 4.05 0.51 0.12 0.00

81,784 13,331 7508 3314 417 95 1

100 10.25 4.49 2.46 0.36 0.13 0.00

3870 397 174 95 14 5 0

100 11.32 4.46 2.55 0.30 0.15 0.00

1594 180 71 41 5 2 0

100 8.97 3.27 2.10 0.28 0.14 0.00

76 7 2 2 0 0 0

100 11.26 3.96 2.71 0.37 0.18 0.00

33 4 1 1 0 0 0

Table 3 Space debris ratio (unit: %) and number (unit: piece) from general algorithm. Post-breakup time

Debris size (m) 0.01–0.05

0h 2h 1 Day 7 Days 30 Days 60 Days 180 Days

0.05–0.1

0.1–0.5

0.5–1.0

P1.0

Ratio

Number

Ratio

Number

Ratio

Number

Ratio

Number

Ratio

Number

100 17.66 9.55 4.08 0.94 0.56 0.45

81,784 14,443 7813 3335 766 455 364

100 10.10 5.09 2.33 0.44 0.23 0.16

3870 391 197 90 17 9 6

100 11.04 5.46 2.63 0.38 0.06 0.00

1594 176 87 42 6 1 0

100 9.21 3.95 1.32 0.00 0.00 0.00

76 7 3 1 0 0 0

100 12.12 3.03 0.00 0.00 0.00 0.00

33 4 1 0 0 0 0

debris are calculated by the faster algorithm. The results see Table 2. It can be seen from the result that 81,784 pieces of debris with 0.01–0.05 m size are induced by the collision breakup, and they will fall into atmosphere rapidly, orbital debris ratio and number are 16.30% and 13,331 pieces, respectively, in post-breakup 2 h, 9.18% and 7508 pieces in 1 day, 4.05% and 3314 pieces in 7 days, 0.51% and 417 pieces in 30 days, and almost no one lives in half year. Orbital debris with bigger than 0.05 m size can be generally surveyed by people nowadays, and 5573 pieces of debris with this size are from this breakup. Orbital debris ratio and number are 10.55% and 588 pieces, respectively, in post-breakup 2 h, 4.45% and 248 pieces in 1 day, 2.49% and 139 pieces in 7 days, 0.34% and 19 pieces in 30 days, and no one lives in half year. For validating the faster algorithm, we simulate the USA 193 satellite breakup event by general algorithm and compare their results. At the time of post-breakup t, in various size ranges, the ratio of orbital debris to breakup debris and the number of orbital debris are calculated by general algorithm. The results see Table 3. It can be seen from the result that 81,784 pieces of debris with 0.01– 0.05 m size are induced by the collision breakup, and orbital debris ratio and number are 17.66% and 14,443 pieces, respectively, in post-breakup 2 h, 9.55% and 7813 pieces in 1 day, 4.08% and 3,335 pieces in 7 days, 0.94% and 766 pieces in 30 days, and very little part lives in half year. 5573 pieces of debris with bigger than 0.05 m size are from this breakup. Orbital debris ratio and number are 10.37%

and 578 pieces, respectively, in post-breakup 2 h, 5.17% and 288 pieces in 1 day, 2.39% and 133 pieces in 7 days, 0.41% and 23 pieces in 30 days, and almost no one lives in half year. By comparing data, we can ﬁnd that calculation results of two algorithms are very close, and diﬀerence between corresponding orbital debris ratios from two tables is less than 1% generally. So the presented faster algorithm is correct and eﬀective. We write the calculation programs of faster algorithm and general algorithm by MATLAB, and perform calculation on a Pentium 4 computer. Table 4 shows calculation times of two algorithms in every size range. It is obvious that calculation time of general algorithm is eﬀected by the number of debris, and rises accordingly. Whereas calculation time of our faster algorithm is independent of debris quantity, and is about 17 s. When debris size is in 0.1– 0.5 m, calculation times of two algorithms are pretty much the same thing, in 0.05–0.1 m, calculation time of faster algorithm is equal to 60.71% of general algorithm, but in Table 4 Calculation times of two algorithms. Debris size (m)

Debris number (piece)

General algorithm (s)

Faster algorithm (s)

P1.0 0.5–1.0 0.1–0.5 0.05–0.1 0.01–0.05

33 76 1594 3870 81,784

<1 <1 13 28 1437

17 17 17 17 17

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0.01–0.05 m, the ratio is only 1.18%. So advantage of faster algorithm is very big in the way of calculation speed and eﬃciency. And fortunately, its calculation precision is held well.

5. Summary NASA standard breakup model and general algorithm are introduced. But this algorithm is diﬃcult when debris quantity is more than hundreds or thousands. So we ﬁnd and present a faster algorithm for calculating debris cloud orbital lifetime and character from spacecraft collision breakup. For validating the faster algorithm, we calculate the USA 193 satellite breakup event and compare with general algorithm. Contrast result indicates that calculation speed and eﬃciency of faster algorithm is very good. When debris size is in 0.01–0.05 m, the faster algorithm is almost a hundred time faster than general algorithm. And at the same time, its calculation precision is held well. The diﬀerence between corresponding orbital debris ratios from two algorithms is less than 1% generally. The essential of faster algorithm is diﬀerential integral method. Area-to-mass ratios, delta velocities and directions of all debris are subdivided into many diﬀerential sections ﬁrstly. Then, number distribution probability and orbital lifetime of debris in every section are calculated. Finally, debris cloud orbital characters in any time of post-breakup are calculated by integral. Subdivision of diﬀerential sections is the key of faster algorithm. The smaller diﬀerential section, the more precise calculation result is, but the worse calculation eﬃciency is. Therefore, Subdivision of diﬀerential sections should take into account the two factors of precision and eﬃciency synthetically. Through lots of cal-

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culations, we ﬁnd the subdivision of ‘‘81020” used by this paper is appropriate. References Bendisch, J., Bunte, K., Klinkrad, H., Krag, H., Martin, C., Sdunnus, H., Walker, R., Wegener, P., Wiedemann, C. The MASTER-2001 model. Adv. Space Res. 34 (5), 959–968, 2004. Johnson, N.L., Krisko, P.H., Liou, J.-C., Anz-Meador, P.D. NASA’s New Breakup Model of EVOLVE 4.0. Adv. Space Res. 28 (9), 1377– 1384, 2001. Li, Y.Y., Shen, H.R., Li, Z. On-orbit collision activities and breakup model, IAC-08-A6.5.11. In: The 59th International Astronautical Congress, Glasgow, Scotland, September 29–October 3, 2008. Liou, J.-C., Hall, D.T., Krisko, P.H., Opiela, J.N. LEGEND – a threedimensional LEO-to-GEO debris evolutionary model. Adv. Space Res. 34 (5), 981–986, 2004. Liou, J.-C. Collision activities in the future orbital debris environment. Adv. Space Res. 38 (9), 2102–2106, 2006. Michael, O., Sebastian, S., Carsten, W., Peter, V., Peter, W., Klinkrad, H. A revised approach for modelling on-orbit fragmentations. In: AIAA2004-5221, AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Providence, Rhode Island, August 16–19, 2004. Michael, O., Sebastian, S., Carsten, W., Peter, W., Clare, M., Klinkrad, H. Upgrade of the MASTER Model, Final Report, Institute of Aerospace Systems, Technische Universitat Braunschweig, European Space Agency, Document ID: M05/MAS-FR, Rev. 1.0, Dated April 26, 2006. Reynolds, R.C., Bade, A., Eichler, P., Jackson, A.A., Krisko, P.H., Matney, M., Kessler, D.J., Anz-Meador, P.D. NASA Standard Breakup Model 1998 Revision. LMSMSS-32532, Lockheed Martin Space Mission Systems and Services, September, 1998. The NASA Orbital Debris Program Oﬃce. Monthly number of objects in earth orbit by object type. Orbital Debris Quarterly News 12(1), 12, 2008. Vallado, D.A. Fundamentals of astrodynamics and applications. Microcosm Press, EL Segundo, CA, 2004. Wikipedia, USA 193, http://en.wikipedia.org/wiki/USA_193#cite_noteNSSDC-craft-1.

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The Science Case for STEP James Overduin, Francis Everitt, John Mester *, Paul Worden Gravity Probe B, Hansen Experimental Physics Laboratory, Stanford University, Stanford, CA 94305, USA Received 13 November 2008; received in revised form 17 February 2009; accepted 19 February 2009

Abstract The Satellite Test of the Equivalence Principle (STEP) will advance experimental limits on violations of Einstein’s Equivalence Principle (EP) from their present sensitivity of 2 parts in 1013 to 1 part in 1018 through multiple comparison of the motions of four pairs of test masses of diﬀerent compositions in an earth-orbiting drag-free satellite. Dimensional arguments suggest that violations, if they exist, should be found in this range, and they are also suggested by leading attempts at uniﬁed theories of fundamental interactions (e.g., string theory) and cosmological theories involving dynamical dark energy. Discovery of a violation would constitute the discovery of a new force of nature and provide a critical signpost toward uniﬁcation. A null result would be just as profound, because it would close oﬀ any possibility of a natural-strength coupling between standard-model ﬁelds and the new light degrees of freedom that such theories generically predict (e.g., dilatons, moduli, quintessence). STEP should thus be seen as the intermediate-scale component of an integrated strategy for fundamental physics experiments that already includes particle accelerators (at the smallest scales) and supernova probes (at the largest). The former may ﬁnd indirect evidence for new ﬁelds via their missing-energy signatures, and the latter may produce direct evidence through changes in cosmological equation of state—but only a gravitational experiment like STEP can go further and reveal how or whether such a ﬁeld couples to the rest of the standard model. It is at once complementary to the other two kinds of tests, and a uniquely powerful probe of fundamental physics in its own right. Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Equivalence principle; Gravitation; Scalar ﬁelds; Dark energy; Drag-free satellite; Superconducting accelerometer

1. Historical overview The Satellite Test of the Equivalence Principle (STEP; Fig. 1) will probe the foundation of Einstein’s theory of general relativity, the (local) equivalence of gravitational and inertial mass—more speciﬁcally called the weak equivalence principle—to unprecedented precision. The equivalence principle (EP) originated in Newton’s clear recognition (1687) of the strange experimental fact that mass fulﬁlls two conceptually independent functions in physics, as both the source of gravitation and the seat of inertia. Einstein’s ‘‘happiest thought” (1907) was the recognition that the equivalence of gravitational and inertial *

Corresponding author. E-mail addresses: [email protected] (J. Overduin), francis @relgyro.stanford.edu (F. Everitt), [email protected] (J. Mester), [email protected] (P. Worden).

mass allows one to locally ‘‘transform away” gravity by moving to the same accelerated frame, regardless of the mass or composition of the falling object. It followed that the phenomenon of gravitation could not depend on any property of matter, but must rather spring from some property of spacetime itself. Einstein identiﬁed the property of spacetime that is responsible for gravitation as its curvature. General relativity, our currently accepted ‘‘geometrical” theory of gravity, thus rests on the validity of the EP. But it is now widely expected that general relativity must break down at some level in order to be united with the other ﬁelds making up the standard model of particle physics. It therefore becomes imperative to test the EP as carefully as possible. Historically, equivalence has been tested in four distinct ways: (1) Galileo’s free-fall method, (2) Newton’s pendulum experiments, (3) Newton’s celestial method (his dazzling insight that moons and planets could be used as test

0273-1177/$36.00 Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2009.02.012

J. Overduin et al. / Advances in Space Research 43 (2009) 1532–1537

Fig. 1. Artists’ impression of the STEP spacecraft.

masses in the ﬁeld of the sun) and (4) Eo¨tvo¨s’ torsion balance. Certain kinds of EP violation can also be constrained by other phenomena such as the polarization of the cosmic microwave background (Ni, 2008). However, the most robust and sensitive EP tests to date have come from approaches (3) and (4). The celestial method now makes use of lunar laser ranging to place limits on the relative difference in acceleration toward the sun of the earth and moon of ð1:0 1:4Þ 1013 (Williams et al., 2004), and modern state-of-the-art torsion-balance experiments give comparable constraints of ð0:3 1:8Þ 1013 (Schlamminger et al., 2008). Both these methods have reached an advanced level of maturity and it is unlikely that they will advance signiﬁcantly beyond the 1013 level in the near term. STEP is conceptually a return to Galileo’s free-fall method, but one that uses a 7000 km high ‘‘tower” that constantly reverses its direction to give a continuous periodic signal, rather than a quadratic 3 s drop (Fig. 2). A free-fall experiment in space has two principal advantages over terrestrial torsion-balance tests: a larger driving acceleration (sourced by the entire mass of the earth) and a quieter ‘‘seismic” environment, particularly if drag-free

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technology is used. These factors alone lead to an increase in sensitivity over current methods of three orders of magnitude each. Comprehensive numerical disturbance analysis (Worden et al., 2001) has established that 20 orbits (1.33 days) of integration time will suﬃce for STEP to improve on existing EP constraints by ﬁve to six orders of magnitude, from 1013 to 1018. There are other proposed experiments to test the EP in space; most notable is MicroSCOPE, a room-temperature mission that will ﬂy two accelerometers with a measurement goal of 1015 . MicroSCOPE was conceived by the ONERA group (part of the STEP collaboration) and is funded by CNES and ESA with a possible launch date of 2011. This paper will focus on STEP, which while not yet approved for ﬂight, has been through Phase A studies sponsored by NASA and ESA, and promises a large improvement in sensitivity. 2. Experimental design The STEP design calls for four pairs of concentric test masses, currently composed of Pt–Ir alloy, Nb and Be in a cyclic condition to eliminate possible sources of systematic error (the total acceleration diﬀerence between A–B, B–C and C–A must be zero for three mass pairs AB, BC and CA). This choice of test-mass materials is not yet ﬁxed, but results from extensive theoretical discussions suggesting that EP violations are likely to be tied to three potential determinative factors that can be connected to a general class of string-inspired models: baryon number, neutron excess and nuclear electrostatic energy [Fig. 3; Damour et al. (1994), Damour (1996), Blaser (2001)]. The test masses are constrained by superconducting magnetic bearings to move in one direction only; they can be perfectly centered by means of gravity gradient signals, thus avoiding the pitfall of most other free-fall methods (unequal initial velocities and times of release). Their accelerations are monitored with very soft magnetic ‘‘springs” coupled to a cryogenic SQUID-based readout system. The SQUIDs are inherited from Gravity Probe B, as are many of the other key STEP technologies, including test-mass caging mechanisms, charge measurement and UV discharge systems, drag-free control algorithms and proportional helium thrusters using boiloﬀ from the dewar as propellant (Fig. 4). Prototypes of key components including the accelerometer are in advanced stages of development. 3. Theoretical motivation

Fig. 2. Compared with Galileo at Pisa, STEP employs a ‘‘tower” not 60 m but 7000 km high, and one constantly reversing its direction to give a continuous periodic signal rather than a quadratic 3 s drop.

Theoretically, the range 1018 K Da=a K 1013 is an extremely interesting one. This can be seen in at least three ways. The simplest argument is a dimensional one. New eﬀects in any theory of quantum gravity must be describable at low energies by an eﬀective ﬁeld theory with new 2 terms like bðm=mQG Þ þ Oðm=mQG Þ where b is a dimensionless coupling parameter not too far from unity and mQG is the quantum-gravity energy scale, which could be any-

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Fig. 3. Test mass choice. A theoretical approach might be to span the largest possible volume in the space of string-inspired ‘‘elementary charges” such as baryon number N þ Z, neutron excess N Z and nuclear electrostatic energy ½/ ZðZ 1Þ, all normalized by atomic mass A (see text). This approach must be balanced against practical issues such as manufacturability, cost, and above all the need to make sure that anything measured is a real eﬀect.

where between the grand uniﬁed theory (GUT) scale mGUT 1016 GeV and the Planck scale mPl 1019 GeV. In a theory combining gravity with the standard model of particle physics, m could plausibly lie anywhere between the mass of an ordinary nucleon ðmnuc 1 GeVÞ and that of the Higgs boson ðmH 100 GeVÞ. With these numbers one ﬁnds that EP-violating eﬀects should appear between ðmnuc =mPl Þ 1019 and ðmH =mGUT Þ 1014 — the range of interest. Adler (2006) has noted that this makes STEP a potential probe of quantum gravity. The dimensional argument, of course, is not decisive. A second approach is then to look at the broad range of speciﬁc theories that are suﬃciently mature to make quantitative predictions for EP violation. There are two main categories. On the high-energy physics side, EP violations occur in many of the leading uniﬁed theories of fundamental interactions, notably string theories based on extra spatial dimensions. In the low-energy limit, these give back classical general relativity with a key diﬀerence: they generically predict the existence of a four-dimensional scalar dilaton partner to Einstein’s tensor graviton, and sev-

eral other gravitational-strength scalar ﬁelds known as moduli. In the early universe, these ﬁelds are naturally of the same order as the gravitational ﬁeld, and some method has to be found to get rid of them in the universe we observe. If they survive, they will couple to standardmodel ﬁelds with the same strength as gravity, producing drastic violations of the EP. One conjecture is that they acquire large masses and thus correspond to very shortrange interactions, but this solution, though widely accepted, entails grave diﬃculties (the Polonyi or ‘‘moduli problem”) because the scalars are so copiously produced in the early universe that their masses should long ago have overclosed the universe, causing it to collapse. Another possibility involves a mechanism whereby a massless ‘‘runaway dilaton” (or moduli) ﬁeld is cosmologically attracted toward values where it almost, but not quite, decouples from matter; this results in EP violations that overlap the range identiﬁed above and might in principle reach the 1012 level (Damour et al., 2002). Similar comments apply to another inﬂuential model, the TeV ‘‘little string” theory (Antoniadis et al., 2001).

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Fig. 4. Cutaway view of the STEP spacecraft (top) with accelerometer detail (bottom; DA, diﬀerential accelerometer; EPS, electrostatic positioning system; SRS, squid readout sensor). Each accelerometer contains two concentric test masses, cylindrical in shape but with dimensions chosen to make them gravitationally ‘‘spherical” to sixth order in mass moments.

The second category of speciﬁc EP-violating theories occurs at the opposite extremes of mass and length, in the ﬁeld of cosmology. The reason, however, is the same: a new ﬁeld is introduced whose properties are such that it should naturally couple with gravitational strength to standard-model ﬁelds, thus inﬂuencing their motion in violation of the EP. The culprit in this case is typically dark energy, a catch-all name for the surprising but observationally unavoidable fact that the expansion of the universe

appears to be undergoing late-time acceleration. Three main explanations have been advanced for this phenomenon: either there is a cosmological constant (whose value is extremely diﬃcult to understand), or general relativity is incorrect on the largest scales—or dark energy is dynamical. Most theories of dynamical dark energy (also known as quintessence) involve one or more species of new, light scalar ﬁelds that could violate the EP (Bean, 2005). The same thing is true of new ﬁelds that may be responsible

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Fig. 5. Investigating nature on all three scales: small, large—and intermediate.

for producing cosmological variations in the values of fundamental constants such as the electromagnetic ﬁne-structure constant a (Dvali and Zaldarriaga, 2002). In all or most of these speciﬁc theories, EP violations typically appear within the STEP range, 1018 K Da=a K 1013 . To understand the reasons for this, it is helpful to look at the third of the arguments alluded to above for regarding this range as a particularly rich and interesting one from a theoretical point of view. This line of reasoning shares some of the robustness of the dimensional argument, in that it makes the fewest possible assumptions beyond the standard model, while at the same time being based upon a convincing body of detailed calculations. Many authors have done work along these lines, with perhaps the best known being that of Carroll (1998) and Chen (2005), which we follow in outline here. Consider the simplest possible new ﬁeld: a scalar / (as motivated by observations of dark energy, or alternatively by the dilaton or supersymmetric moduli ﬁelds of high-energy uniﬁed theories such as string theory). Absent some protective symmetry (whose existence would itself require explanation), this new ﬁeld / couples to standard-model ﬁelds via dimensionless coupling constants bk (one for each standard-model ﬁeld) with values not too far from unity. Detailed calculations within the standard model (modiﬁed only to incorporate /) show that these couplings are tightly constrained by existing limits on violations of the EP. The current bound of order Da=a < 1012 translates directly into a requirement that the dominant coupling factor (the one associated with the gauge ﬁeld of quantum chromodynamics or QCD) cannot be larger than bQCD < 106 . This is very small for a dimensionless coupling constant, though one can plausibly ‘‘manufacture” dimensionless quantities of this size (e.g., a2 =16p), and many theorists would judge that anything smaller is almost certainly zero. Now STEP will be sensitive to violations as small as 1018 . If none are detected at this

level, then the corresponding upper bounds on bQCD go down like the square root of Da=a; i.e., to bQCD < 109 , which is no longer a natural coupling constant by any current stretch of the imagination. For perspective, recall the analogous ‘‘strong CP” problem in QCD, where a dimensionless quantity of order 108 is deemed so unnatural that a new particle, the axion, must be invoked to drive it toward zero. This argument does not say that EP violations inside the STEP range are inevitable; rather it suggests that violations outside that range would be so unnaturally ﬁnetuned as to not be worth looking for. As Ed Witten has stated, ‘‘It would be surprising if / exists and would not be detected in an experiment that improves bounds on EP violations by 6 orders of magnitude” (Witten, 2000). Only a space test of the EP has the power to force us to this conclusion. The fundamental nature of the EP makes such a test a win–win proposition, regardless of whether violations are actually detected. A positive detection would be equivalent to the discovery of a new force of nature, and our ﬁrst signpost toward uniﬁcation. A null result would imply either that no such ﬁeld exists, or that there is some deep new symmetry that prevents its being coupled to SM ﬁelds. A historical parallel to a null result might be the Michelson–Morley experiment, which reshaped physics because it found nothing. The ‘‘nothing” ﬁnally forced physicists to accept the fundamentally diﬀerent nature of light, at the cost of a radical revision of their concepts of space and time. A non-detection of EP violations at the 1018 level would strongly suggest that gravity is so fundamentally diﬀerent from the other forces that a similarly radical rethinking will be necessary to accommodate it within the same theoretical framework as the SM based on quantum ﬁeld theory. STEP should be seen as the integral intermediate-scale element of a concerted strategy for fundamental physics

J. Overduin et al. / Advances in Space Research 43 (2009) 1532–1537

experiments that already includes high-energy particle accelerators (at the smallest scales) and cosmological probes (at the largest scales), as suggested in Fig. 5. Accelerators such as the Large Hadron Collider (LHC) may provide indirect evidence for the existence of new ﬁelds via their missing-energy signatures. Astronomical observatories such as the SuperNova Acceleration Probe (SNAP) may produce direct evidence of a quintessence-type cosmological ﬁeld through its bulk equation of state. But only a gravitational experiment such as STEP can go further and reveal how or whether that ﬁeld couples to the rest of the standard model. It is at once complementary to the other two kinds of tests, and a uniquely powerful probe of fundamental physics in its own right. Acknowledgement Thanks go to Sean Carroll for enlightening discussions. References Adler, R. Gravity, in: Fraser, G. (Ed.), The New Physics for the TwentyFirst Century. Cambridge University Press, Cambridge, pp. 41–68, 2006. Antoniadis, I., Dimopoulos, S., Giveon, A. Little string theory at a TeV. J. High Energy Phys. 5, 055-1–055-22, 2001. Bean, R., Carroll, S., Trodden, M., Insights into dark energy: interplay between theory and observation (white paper submitted to the Dark Energy Task Force). arXiv:astro-ph/0510059, 2005.

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Blaser, J.P. Test mass material selection for equivalence principle experiments. Class. Quant. Grav. 18, 2515–2520, 2001. Carroll, S.M. Quintessence and the rest of the world: suppressing longrange interactions. Phys. Rev. Lett. 81, 3067–3070, 1998. Chen, J. Probing scalar couplings through tests of the equivalence principle, Ph.D. thesis UMI-31-95001 (University of Chicago), 34 pp., 2005. Damour, T. Testing the equivalence principle: why and how? Class. Quant. Grav. 13, A33–A41, 1996. Damour, T., Blaser, J.P. Optimizing the choice of materials in equivalence principle experiments, in: Tran Thanh Van, F., Fontaine, G., Hinds, E. (Eds.), Particle Astrophysics, Atomic Physics and Gravitation (Editions Frontieres), pp. 433–440, 1994. Damour, T., Piazza, F., Veneziano, G. Runaway dilaton and equivalence principle violations. Phys. Rev. Lett. 89, 081601-1–081601-4, 2002. Dvali, G., Zaldarriaga, M. Changing a with time: implications for ﬁfthforce-type experiments and quintessence. Phys. Rev. Lett. 88, 0913031–091303-4, 2002. Ni, W.-T. From equivalence principles to cosmology: cosmic polarization rotation, CMB observation, neutrino number asymmetry, Lorentz invariance and CPT. Prog. Theor. Phys. Suppl. 172, 49–60, 2008. Schlamminger, S., Choi, K.-Y., Wagner, T.A., et al. Test of the equivalence principle using a rotating torsion balance. Phys. Rev. Lett. 100, 041101-1–041101-4, 2008. Williams, J.G., Turyshev, S.F., Boggs, D.H. Progress in lunar laser ranging tests of relativistic gravity. Phys. Rev. Lett. 93, 261101-1– 261101-4, 2004. Witten, E. The cosmological constant from the viewpoint of string theory, in: Cline, D.B. (Ed.), Sources and Detection of Dark Matter and Dark Energy in the Universe. World Scientiﬁc, pp. 27–36, 2000. Worden, P., Mester, J., Torii, R. STEP error model development. Class. Quant. Grav. 18, 2543–2550, 2001.

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Advances in Space Research 43 (2009) 1538–1544 www.elsevier.com/locate/asr

Pioneer 10 Doppler data analysis: Disentangling periodic and secular anomalies A. Levy a, B. Christophe a,*, P. Be´rio b, G. Me´tris b, J.-M. Courty c, S. Reynaud c b

a ONERA/DMPH, 29 av. Division Leclerc, F-92322 Chatillon, France Geoscience Azur, Universite´ Nice Sophia-Antipolis, OCA, Avenue Copernic F-06130 Grasse, France c LKB, UPMC, Case 74, CNRS, ENS, F-75252 Paris cedex 05, France

Received 15 September 2008; received in revised form 12 December 2008; accepted 11 January 2009

Abstract This paper reports the results of an analysis of the Doppler tracking data of Pioneer probes which did show an anomalous behaviour. A software has been developed for the sake of performing a data analysis as independent as possible from that of Anderson et al. [Anderson, J., Laing, P.A., Lau, E.L., Liu, A.S., Nieto, M.M., Turyshev, S.G. Study of the anomalous acceleration of Pioneer 10 and 11. Phys. Rev. D 65, 082004, 2002], using the same data set. A ﬁrst output of this new analysis is a conﬁrmation of the existence of a secular anomaly with an amplitude about 0.8 nm s2 compatible with that reported by Anderson et al. A second output is the study of periodic variations of the anomaly, which we characterize as functions of the azimuthal angle u deﬁned by the directions Sun–Earth Antenna and Sun-Pioneer. An improved ﬁt is obtained with periodic variations written as the sum of a secular acceleration and two sinusoids of the angles u and 2u. The tests which have been performed for assessing the robustness of these results are presented. Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Pioneer anomaly; Doppler tracking data; Interplanetary trajectography

1. Introduction The Pioneer 10 and 11 spacecrafts performed accurate celestial mechanics experiment after their encounters with Jupiter and Saturn. Doppler tracking by the NASA Deep Space Network (Moyer, 2000) of the two spacecrafts which followed escape orbits to opposite ends of the solar system revealed an anomalous behaviour which remains imperfectly understood. Precisely, the tracking data do not meet the expectations drawn from standard gravity force law and a better ﬁt is obtained by adding a constant Sun-ward acceleration (Anderson et al., 1998). This anomalous acceleration, now referred to as Pioneer anomaly, has a similar magnitude for the two spacecrafts. The value reported by *

Corresponding author. Tel.: +33 1 46 73 49 35; fax: +33 1 46 73 48 24. E-mail addresses: [email protected] (A. Levy, B. Christophe), [email protected] (P. Be´rio, G. Me´tris), [email protected] (J.-M. Courty, S. Reynaud).

the JPL team who discovered the anomaly is 0:874 0:133 nm s2 (Anderson et al., 2002). The extensive analysis by the JPL team has been published after years of cross-checks (Anderson et al., 2002). The presence of an anomaly and its magnitude have been conﬁrmed by diﬀerent analysis software (Markwardt, 2002; Olsen, 2007). A number of mechanisms have been considered as attempts of explanations of the anomaly as a systematic eﬀect generated by the spacecraft itself or its environment (see as an example Nieto et al., 2005) but they have not led to a satisfactory understanding to date. If conﬁrmed, the Pioneer signal might reveal an anomalous behaviour of gravity at scales of the order of the size of the solar system and thus have a strong impact on fundamental physics, astrophysics and cosmology. It is therefore important to explore all its facets in the wider context of navigation and test of the gravity law in the solar system (see for example Bertolami, 2006; Reynaud, accepted for publication, and references therein).

0273-1177/$36.00 Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2009.01.003

A. Levy et al. / Advances in Space Research 43 (2009) 1538–1544

Crucial informations are expected to come out from the re-analysis of Pioneer data (Nieto and Anderson, 2005; Turyshev et al., 2006). An international collaboration has been built to this aim within the frame of International Space Science Institute (ISSI, 2006–2008). The Pioneer data which had been analysed by the Anderson team (Anderson et al., 2002) have been made available by Slava Turyshev (JPL) in the framework of this collaboration. They consist in Orbit Data Files (ODF) which contain in particular Pioneer 10 Doppler data from November 30th, 1986 to July 20th, 1998. The aim of the present paper is to report some results of the analysis of these data performed by a collaboration between three groups at ONERA, OCA and LKB within the ‘‘Groupe Anomalie Pioneer” (Groupe Anomalie Pioneer, 2006). A dedicated software called ODYSSEY has been developed to this purpose. The ﬁrst result is to conﬁrm the existence and magnitude of the anomalous secular acceleration reported by Anderson et al., using diﬀerent and as independent as possible tools. The main motivation of the present paper is to study the periodic variations of the anomaly, which are known to exist besides the secular Pioneer anomaly. The presence of diurnal and seasonal variations in the residuals has been discussed by Anderson et al. (2002). We may emphasize that such modulated anomalies are unlikely to be due to anything related to the spacecraft or its environment. Anderson et al. (2002) have proposed that they are due to modeling errors such as errors in the Earth’s ephemeris, the orientation of the Earth’s spin axis or the station’s coordinates. However, these parameters are strongly constrained by other observational methods and it seems diﬃcult to change them enough to explain the periodic anomaly. We will show here that this periodic anomaly can be at least partly represented in terms of a modulation of the Doppler signal as a function of a unique azimuthal angle having a physical meaning. 2. Development of the ODYSSEY software The Pioneer 10/11 spacecrafts were tracked by the Deep Space Network (DSN) antennas. A S-Band signal at about 2.11 GHz is emitted at time t1 by a DSN antenna and received onboard the spacecraft at time t2 . The frequency is multiplied by a constant ratio of 240/221 by a transponder and sent back to a ground antenna where it was received at time t3 . The Doppler shift, that is the diﬀerence between the up- and down-frequencies, is called a 2-way observable if the two antennas are the same, a 3-way observable if they are diﬀerent. In fact, the ODF observable is the average of the Doppler shift over a time span called compression interval (Moyer, 2000). The ODF format, described in DSN (2000), also contains the compression time, the date of the middle of the compression interval, the emitted frequency and the receiving and transmitting DSN antenna identiﬁers.

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To analyse the ODF data, a software called ODYSSEY has been developed at the Observatoire de la Coˆte d’Azur within a collaboration with Onera. ODYSSEY stands for ‘‘Orbit Determination and phYsical Studies in the Solar Environment Yonder”. One of the motivation for the development of this software is the simulation of the expected eﬀects of the Solar System Odyssey project. It is basically an interplanetary trajectory determination software. It performs numerical integration in rectangular coordinates of dynamical equations to propagate the position and velocity of the spacecraft and variational equations to propagate the sensitivity of the position and velocity with respect to the initial conditions of position and velocity and other parameters to be ﬁtted. The numerical integration uses the Adams–Moulton–Cowell algorithm (Henrici, 1962) at an order 10. The use of this algorithm enables the direct integration of second order equations. The values of the parameters to be estimated are obtained through a best ﬁt procedure. The diﬀerence between measured observables O and the calculated ones C, which depend on the parameters E to be ﬁtted, is linearized in terms of variations of E. For N values of the observable, this corresponds to a linear system of N equations with as many variables as there are terms in E. This system is then solved in ODYSSEY with classical iterative least squares analysis. Maneuvers are taken into account as increments of velocity dV along the three directions. The dates of the maneuvers are provided by JPL and their amplitudes estimated as parameters in the best ﬁt analysis. Constraints are imposed on the maximum values of the estimated dV in non-radial directions. The dynamical model to compute the motion of the spacecraft includes gravitational attraction by the main bodies of the solar system and direct radiation pressure. The motion of the spacecraft is computed using non-relativistic gravitational equations, which are known to be sufﬁcient at the considered level of accuracy (Markwardt, 2002). For the solar radiation pressure we use the same model as in Anderson et al. (2002) with the reﬂectivity coefﬁcient k r ¼ 1:77, the mass m ¼ 259 kg of the spacecraft and a cross-sectional area S ¼ 5:9 m2 (orientation of the antenna supposed here to be constant). The accuracy of this model is not critical because the acceleration due to the solar radiation pressure has decreased to less than 0:2 nm s2 for Pioneer 10 at the considered distances. Other known standard perturbations have been found to be negligible for Pioneer at the considered period (Anderson et al., 2002). In its present status, the data analysis does not take into account the detailed thermal models of the spacecraft, currently under study by diﬀerent groups. These models are expected to produce a slowly evoluting radiation force due to heat dissipation from the Radioisotope Thermoelectric Generators (RTG); this force should appear as a part of the anomalous secular acceleration to be found below.

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Improved analysis of this important issue will only be possible when accurate enough models of a time dependent radiative budget will be available. The dynamical equations are integrated in the Barycentric Celestial Reference System (BCRS); The reference time scale is the Barycentric Coordinate Time (TCB). Positions of terrestrial stations are expressed in the International Terrestrial Reference System (ITRS); the reference time scale is the Coordinated Universal Time (UTC). The transition between ITRS and BCRS on the one hand, and TCB and UTC on the other hand, are performed according to 2003 International Earth Rotation Service (IERS) conventions. The positions of celestial bodies are obtained from DE 405 ephemeris from JPL in BCRS with a reference time scale which is refered to as ‘‘T eph ” and which is similar to the Barycentric Dynamical Time (Standish, 1998). Special eﬀorts have been devoted in the development of ODYSSEY for handling the calculation of the ODF observable. In a ﬁrst step, the perturbations of the round-trip light time are not taken into account. The ODF observable, that is the average of the Doppler shift over the compression time, is computed through a numerical approximation using the 4-points Simpson method. For the available Pioneer data, the compression time does not exceed 1980 s. The accuracy of the 4-points Simpson method in this case is evaluated to be 0.4 mHz. The instantaneous Doppler shift is calculated in terms of velocities of the endpoints, evaluated at the event times t1 , t2 and t3 (Markwardt, 2002). As only t3 is provided in ODF, t2 and t1 have to be determined, which is done iteratively using the light time equation r23 ðt2 ; t3 Þ d‘23 ðt2 ; t3 Þ; c r12 ðt1 ; t2 Þ d‘12 ðt1 ; t2 Þ t1 ¼ t2 c

t2 ¼ t3

ð1Þ

where d‘12 and d‘23 are, respectively, the Shapiro time delay aﬀecting the up- and down-link signals. The spin frequency involved in Doppler shift estimation is provided by JPL. The ODF observable comes from JPL already corrected for data after the 17th of July 1990. Before this date, the spin correction is done by our software, taking the closest available value in the provided time series. As the maximal diﬀerence between two successive spin values in the ﬁle is 0.2 mHz, the accuracy of this method is considered as suﬃcient. In the second step of the computation of the observable, the perturbations which aﬀect the propagation of the tracking signal are taken into account. The Shapiro time delay and the solar corona eﬀect are modeled as in Anderson et al. (2002). For the determination of the electron density necessary to compute ionospheric eﬀect, two models have been implemented, the International Reference Ionosphere (IRI) 2007 (Bilitza, 2001) and the Parametrized Ionospheric Model (PIM) (Daniell et al., 1995). Two models have been implemented in ODYSSEY for the mapping functions of the tropospheric eﬀect, the Niell

Mapping Functions (NMF) (Niell, 1996) and the Global Mapping Function (GMF) (Boehm et al., 2006). All these perturbations are modeled as delays aﬀecting the signal (except for the Shapiro delay, their eﬀect in Eq. (1) is however negligible). They are written as added propagation lengths, d‘12 and d‘23 for the up- and down-links. Their eﬀect on the ODF observable is therefore given by the diﬀerence of these quantities between the ‘‘start” and ‘‘stop” of the compression interval (Moyer, 2000). 3. Conﬁrmation of the existence of a constant anomaly Our ﬁrst aim was to study the secular Pioneer anomaly reported by Anderson et al. (2002). To this aim, we performed a best-ﬁt with a constant anomalous acceleration aP exerted on the probe. The acceleration was centered on a point chosen at the Sun, the Solar System Barycentre (SSB) or the Earth. The results obtained with these 3 possibilities were not distinguishable from each other, due to the large distance of the probe to the center of the solar system. From now on, we consider the center to be the Sun. The initial conditions as well as the three components of each maneuver are also ﬁtted. The components of the maneuvers are ﬁtted in the RTN (Radial Transverse Normal) frame and the transverse and normal amplitudes are constrained to be inferior to 0.2 m s1. The IRI 2007 model for ionospheric correction and GMF for tropospheric correction are used. Points with an elevation inferior to 20° are rejected so as to limit the eﬀect of imperfections of atmospherical models. Outliers are also rejected when their difference with the expectation exceeds 100 Hz at the ﬁrst iteration and 6r at the following iterations with r the standard deviation of the residuals at this iteration. The analysis performed with the software ODYSSEY conﬁrms that a better ﬁt is obtained with a constant sunward acceleration. The value estimated by ODYSSEY for the anomalous acceleration is aP ¼ 0:84 0:01 nm s2 with the formal error given at 1r. The postﬁt residuals show a standard deviation of 9.8 mHz, which is largely improved with respect to a ﬁt without aP . These residuals are shown in Fig. 1. In Fig. 2 is given another representation agreeing with Fig. 8 of Anderson et al. (2002). This ﬁgure is reconstructed by the following procedure: the postﬁt residuals are _ opt opt ; aP Þ where X opt _ opt expressed as O CðX opt 0 ;X0 ;M 0 , X0 opt and M are the optimized values for initial position, initial velocity and maneuver amplitudes; then Fig. 2 represents _ opt opt ; 0Þ where the anomalous the quantity O CðX opt 0 ;X0 ;M acceleration has been nulliﬁed; this representation highlights the need of the constant acceleration to reduce the residuals. It can be emphasized that the level of the residuals in Fig. 1 is higher than the measurement noise. It is also clear on the ﬁgure that the postﬁt residuals do not correspond to a white gaussian noise. In order to highlight the existence of systematic structures in the residuals, we zoom on a time interval corresponding to the period from 23 November

60

Residuals (mHz)

40

20

0

−20

−40

−60 1986

1988

1990

1992 1994 Date (year)

1996

1998

2000

Fig. 1. Best ﬁt residuals of the Doppler tracking data of Pioneer 10 with an anomalous acceleration aP .

400

postﬁt residuals. As the Pioneer data points are not evenly distributed, the commonly used Fourier transform is not appropriate for the analysis. We have used the software SparSpec which estimates frequency distributions in time series (Bourguignon, 2007). The result of this spectral analysis is shown in Fig. 5. The presence of signiﬁcant periodic terms is clear at the periods measured with respect to a day = 86,400 s:

300 200 100 0 −100 −200 −300 −400 0

500

1000

1500

2000

2500

3000

1996 to 23 December 1996 where Pioneer 10 was on opposition (Sun, Earth and Pioneer 10 aligned in this order). The data set is thus less aﬀected by solar plasma and it shows daily variations of the residuals (see Fig. 3). This ﬁgure clearly shows the existence of daily variations in the residuals, which are emphasized by distinguishing the data according to diﬀerent couples of emission/reception antennas. Diﬀerent symbols or colors refer to diﬀerent couples of stations, with M standing for Madrid (DSN antenna 63), G for Goldstone (DSN antenna 14), C for Canberra (DSN antenna 43). The full line is a daily sinusoid ﬁtted to the residuals of the dominant data (M,M). 4. Study of the periodic variations of the Pioneer anomaly In order to characterize the periodic variations of the residuals, we have performed a spectral analysis of the

f1 ¼ 0:9974 0:0004 day 1 f2 ¼ ð0:9972 0:0004Þ day 2 f3 ¼ 189 32 day: As 0.9972 day = 1.0 sidereal day, these periods are consistent with variations on one sidereal day, half a sidereal day, and half a year. The presence of diurnal and seasonal variations in the residuals has also been reported by Anderson et al. (2002). It has to be noted that these speciﬁc periods are unlikely to be due to anything related to the spacecraft or its environment. Errors in the atmospherical models would induce daily variations. As these eﬀects depend on the conditions at the stations, such errors would be expected at solar day rather than at sidereal day. Anderson et al. (2002) propose modeling errors such as in the case of Earth’s ephemeris, the orientation of Earth’s spin axis or the station’s coordinates. However, these parameters are strongly constrained by other observational methods and it seems diﬃcult to change them enough to explain the periodic anomaly. The main motivation of the present paper is to test an alternative explanation where some perturbation would modify the propagation of the tracking signal along the path from the Earth antenna and the spacecraft. The idea is to represent such a perturbation, whatever its origin, as

A. Levy et al. / Advances in Space Research 43 (2009) 1538–1544

a function of the angle u deﬁned as the diﬀerence between the Earth Antenna (A) azimuthal angle and the Pioneer (P) azimuthal angle: u ¼ uP uA (see Fig. 4). The main interest of this geometrical model is that it should simultaneously account for the orbital movement of the Earth around the Sun and the diurnal rotation of the Earth. As this perturbation is supposed to be periodic, it will be represented by a few Fourier coeﬃcients X Df ¼ tn ðcosðnuu Þ þ cosðnud ÞÞ þ t0n ðsinðnuu Þ þ sinðnud ÞÞ n

ð2Þ

Here uu and ud are the angles u evaluated on the upand down-links. Several tests have been performed with diﬀerent numbers of coeﬃcients tn and t0n ﬁtted in addition to the constant anomalous acceleration model. The best results have been obtained with superpositions of functions with n ¼ 1 and 2. The addition of higher order coeﬃcients has not modiﬁed signiﬁcantly the residuals. To summarize, the new ﬁt is identical to the previous one but for the addition of the following modiﬁcation of the Doppler signal:

3.5 3 2.5

6 months 0.5 day

2

1 day

1.5 1 0.5 0 −2 10

0

2

10

4

10 Period (day)

10

Fig. 5. SparSpec analysis of the residuals from the ﬁt with a constant acceleration.

Table 1 Results of the best ﬁt with periodic terms in the signal, with two diﬀerent ionospheric models.

Df ¼ t1 ðcosðuu Þ þ cosðud ÞÞ þ t01 ðsinðuu Þ þ sinðud ÞÞ þ t2 ðcosð2uu Þ þ cosð2ud ÞÞ þ t02 ðsinð2uu Þ þ sinð2ud ÞÞ

4

Amplitude (mHz)

1542

ð3Þ

This model results in a spectacular improvement of the best ﬁt residuals, with the standard deviation reduced by a factor of the order of two. Precisely, the standard deviation of the residuals which was 9.8 mHz without the periodic terms has been decreased to 5.5 mHz. The values of the ﬁtted anomalous parameters are reported in Table 1, with two diﬀerent ionospheric models. The robustness of the results with respect to other modiﬁcations of the procedure is evaluated by the few tests which follow. Using the IRI model and selecting data according to minimal elevation or outlier deﬁnition, we obtain the results shown in Tables 2 and 3. These tables show that the best ﬁt results are robust enough, except for the amplitude t1 , when details of the procedure are changed. In particular, the anomalous secular acceleration is not very sensitive to these changes and it remains roughly

TEC model

IRI 2007

PIM

aP (pms2 ) t1 (mHz) t01 (mHz) t2 (mHz) t02 (mHz) Residuals (mHz)

836 1 124:3 9:3 125:3 0:6 2:7 0:2 4:8 0:1 5.5

836 1 141:6 9:3 127:3 0:6 3:0 0:2 4:9 0:1 5.5

Table 2 Results of the best ﬁt with periodic terms in the signal, when varying the minimal evaluation for the considered data. The initial number of points is 19,805. Min. elevation 2

aP (pm s ) t1 (mHz) t01 (mHz) t2 (mHz) t02 (mHz) No. of eliminated points Residuals (mHz)

0°

10°

20°

30°

844 1 86:4 9:3 111:8 0:6 2:1 0:2 4:7 0:1 789

843 1 98:0 9:2 119:5 0:7 2:3 0:2 4:7 0:1 911

836 1 124 9:3 125 0:6 2:7 0:2 4:8 0:1 2996

826 1 95 10:5 127 0:6 2:1 0:2 4:6 0:1 8584

6.0

5.9

5.5

5.1

Table 3 Results of the best ﬁt with periodic terms in the signal, when varying the outlier criterion at Nr. The initial number of points is 19,805.

Fig. 4. Deﬁnition of the angle u.

Outliers at N r

N=3

N=6

N=8

aP (pm s2 ) t1 (mHz) t01 (mHz) t2 (mHz) t02 (mHz) No. of eliminated points Residuals (mHz)

806 1 197 7:2 126 0:5 3:4 0:1 5:0 0:09 3879 4.16

836 1 124 9:3 125 0:6 2:7 0:2 4:7 0:1 2996 5.5

848 1 102 10:0 124 0:7 2:6 0:2 4:7 0:1 2933 5.9

A. Levy et al. / Advances in Space Research 43 (2009) 1538–1544

5. Conclusion In the present paper, we have reported the ﬁrst results of our re-analysis of the Pioneer 10 Doppler data for the 1986 to 1998 time span. The data are the same that were already analysed in Anderson et al. (2002), Markwardt (2002), and Olsen (2007) but they have been dealt with by using a dedicated software ODYSSEY developed to this purpose. The improvement of the data ﬁt with a constant anomalous acceleration exerted on Pioneer 10 has been conﬁrmed by this new data analysis. Its magnitude is compatible with that reported by Anderson et al. The paper has then been focused on the study of periodic terms in the residuals, which were already noticed in

4 3.5 3 Amplitude (mHz)

the same that in the previous best analysis without modulated terms. For the parameters other than t1 , the uncertainty in the determination has to be estimated from the variations of the results when the details are changed, and certainly not from the formal errors given for a given ﬁt which are smaller. For the amplitude t1 , which is clearly less stable than the other parameters, it is better to consider that it is not determined by the best ﬁt procedure. This feature can plausibly be understood from the following argument. The direction from the Sun to Pioneer 10 varies only slowly, because of the large distance of the probe. It follows that the periodic terms contain essentially variations at one year and one sidereal day for n ¼ 1, at half a year and half a sidereal day for n ¼ 2. The ﬁt of the initial conditions also induces terms at the same periods. In particular, a change of the initial conditions may easily produce variations masking modulated terms (Courty, accepted for publication; Jaekel and Reynaud, 2005). This is probably the reason why the yearly period, which corresponds to a large potential amplitude, was in fact not detected in the spectral analysis of the best ﬁt residuals with only a secular anomaly. Now the best ﬁt with modulated terms tell us a diﬀerent story. The modulated terms are looked for in a dedicated best ﬁt procedure and they are unambiguously found to diﬀer from zero. The values of aP , t01 , t2 and t02 seem to have robust estimates, while t1 is less stable. The analysis of the correlation shows a large value between t1 and the initial conditions (0.814 compared to 1 for a total correlation). There is an interdependance between the estimated values of t1 and the initial conditions. This large correlation reﬂects the phenomenon described in the previous paragraph explaining the instability of t1 estimation. An even more impressive demonstration of the improvement of the data analysis drawn by the inclusion of modulated terms comes from the spectral analysis of the residuals. This spectral analysis is represented in Fig. 6 for the best ﬁt with modulated terms. It is drawn intentionally at the same scale as Fig. 5 so that one can easily notice the global reduction of the main peaks in the spectrum as well as all the secondary peaks.

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2.5 2 1.5 1 0.5 0 −2 10

0

10

2

10 Period (day)

4

10

Fig. 6. SparSpec analysis of the residuals from the ﬁt with a constant acceleration and periodic terms.

previous studies and are clearly revealed using spectral analysis. The main new result of the paper is that a large part of these diurnal and seasonal anomalies may be captured in a simple geometrical model where the light time on the tracking path is modiﬁed in a manner depending only on the azimuthal angle u between the Sun–Earth and Sun-probe lines. This geometrical model could represent in a simple way the physical eﬀects expected on light propagation in some metric extensions of general relativity which have been studied as potential candidates for the explanation of the secular Pioneer anomaly (Jaekel and Reynaud, 2005, 2006). The search for modulated anomalies might even be used to distinguish those candidates from other which aﬀect much less the propagation of tracking signals (Moﬀat, 2006). However, the results of the paper cannot be considered as pointing to a particular possible explanation of the anomaly. At the present stage of the data analysis, similar eﬀects could for example be obtained through a mismodeling of the solar corona model. Anyway, considering the modulated anomalies has allowed us to reduce by a factor of the order of two the standard deviation of the residuals. The results of the best ﬁt have been found to be robust enough with respect to changes in the tropospheric or ionospheric model, as well as variations of the details of the best ﬁt procedure. The most impressive output of the new analysis is given by the SparSpec analysis of the best ﬁt residuals which shows a reduction of all periodic structures which were present in the residuals of the best ﬁt without modulated terms. It is worth emphasizing that this has been done by considering only a function of the geometrical angle u, and not by using diﬀerent explanations for the diurnal and seasonal anomalies. This suggests that the new analysis constitutes a richer characterization of the Pioneer data, now involving not only a secular acceleration but also modulated terms, which will have to be compared with any,

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existing as well as future, possible explanation of the anomaly. Acknowledgements This work has beneﬁted of discussions with a number of people involved in the international collaborations devoted to the investigation of the Pioneer data. Special thanks are due to S.G. Turyshev (NASA JPL) for having led the ISSI team (ISSI, 2006–2008) during which the ODF used in the seminal paper (Anderson et al., 2002) were made available to the members of the collaboration. In particular, the development of ODYSSEY software has beneﬁted from fruitful discussions with Pierre Exertier (OCA). We are also grateful for discussions with the members of this collaboration and especially with P. Touboul and B. Foulon (ONERA) and the members of the french collaboration GAP (Groupe Anomalie Pioneer) (Groupe Anomalie Pioneer, 2006). Another special thanks is due to CNES for its support of the GAP. References Anderson, J.D., Laing, P.A., Lau, E.L., Liu, A.S., Nieto, M.M., Turyshev, S.G. Study of the anomalous acceleration of Pioneer 10 and 11. Phys. Rev. D 65, 082004, 2002. Anderson, J.D, Laing, P.A., Lau, E.L., Liu, A.S., Nieto, M.M., Turyshev, S.G. Indication, from Pioneer 10/11, Galileo, and Ulysses data, of an apparent anomalous, weak, long-range acceleration. Phys. Rev. Lett. 81, 2858–2861, 1998. Bertolami, O., Paramos, J., Turyshev, S.G. General theory of relativity: will it survive the next decade? in: Dittus, H. et al. (Eds.), Lasers, Clocks, and Dragfree: Technologies for Future Exploration in Space and Tests of Gravity, pp. 27–67, 2006. Bilitza, D. International reference ionosphere 2000. Radio Sci. 36 (2), 261– 275, 2001. Boehm, J., Niell, A., Tregoning, P., Schuh, H. Global mapping function (GMF): a new empirical mapping function based on numerical weather model data. Geophys. Res. Lett. 33, L07304, 2006. Bourguignon, S., Carfantan, H., Bohm, T. Astronomy and astrophysics, vol. 462, pp. 379–387, 2007; the SparSpec software is available at http://www.ast.obs-mip.fr/Softwares.

Courty, J.-M., Reynaud, S., Levy, A., Christophe, B. Simulation of ambiguity eﬀects in Doppler tracking of Pioneer probes, Space Sci. Rev., accepted for publication. Daniell, R.E., Anderson, D.N., Fox, M.W., et al. Parametrized ionospheric model-A global ionospheric parametrization based on ﬁrst principles models. Radio Sci. 30 (5), 1499–1510, 1995. DSN Tracking System Interfaces, Orbit Data File Interface, TRK-2-18, Document 820-013 Deep Space Mission Systems, External Interface Speciﬁcation, JPL D-16765, Jet Propulsion Laboratory, Pasadena, CA, 2000. Groupe Anomalie Pioneer, French Collaboration on Pioneer Anomaly regrouping LKB, ONERA, OCA, IOTA and SYRTE laboratories (from 2006), http://www.lkb.ens.fr/-GAP-. Henrici, P. Discrete Variable Methods in Ordinary Diﬀerential Equations. Wiley, 1962. International Collaboration on Pioneer Anomaly at ISSI, Bern, Switzerland (2006–2008), http://www.issi.unibe.ch/teams/Pioneer/. Jaekel, M.-T., Reynaud, S. Post-Einsteinian tests of linearized gravitation. Classical Quantum Gravity 22, 2135–2158, 2005. Jaekel, M.-T., Reynaud, S. Radar ranging and Doppler tracking in postEinsteinian metric theories of gravity. Classical Quantum Gravity 23, 7561–7579, 2006. Markwardt, C. Independent Conﬁrmation of the Pioneer 10 Anomalous Acceleration, arXiv:gr-qc/0208046v1, 2002. Moﬀat, J.W. Time delay predictions in a modiﬁed gravity theory. Classical Quantum Gravity 23, 6767–6771, 2006. Moyer, T.D. Formulation for observed and computed values of deep space network data types for navigation. Deep Space Commun. Navigation Ser., 2000. Niell, A.E. Global mapping functions for the atmosphere delay at radio wavelengths. J. Geophys. Res. 101 (B2), 3227–3246, 1996. Nieto, M.M., Turyshev, S.G., Anderson, J.D. Directly measured limit on the interplanetary matter density from Pioneer 10 and 11. Phys. Lett. B 613, 11–19, 2005. Nieto, M.M., Anderson, J.D. Using early data to illuminate the Pioneer anomaly. Classical Quantum Gravity 22, 5343–5354, 2005. Olsen, O. The constancy of the Pioneer anomalous acceleration. A&A 463, 393–397, 2007. Reynaud, S., Jaekel, M.-T. Tests of general relativity in the solar system, in: Proceedings of the Varenna school on atom optics and space physics, Societa` Italiana di Fisica, arXiv:0801.3407, accepted for publication. Standish, E.M. Time scales in the JPL and CfA ephemerides. A&A 336, 381–384, 1998. Turyshev, S.G., Toth, V.T., Kellogg, L.R., et al. The study of the Pioneer anomaly: new data and objectives for new investigation. Int. J. Mod. Phys. D 15, 1–56, 2006.

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Advances in Space Research 43 (2009) 1545–1554 www.elsevier.com/locate/asr

Impact of emissions from anthropogenic sources on satellite-derived reﬂectance K.V.S. Badarinath, Anu Rani Sharma *, Shailesh Kumar Kharol Atmospheric Science Section, National Remote Sensing Centre, Department of Space-Government of India, Balanagar, Hyderabad 500 037, India Received 29 July 2008; received in revised form 21 October 2008; accepted 18 January 2009

Abstract The present study deals with the impact of extensive anthropogenic activities associated with festivities and agricultural crop residues burning in the Indo-Gangetic Plains (IGP) on satellite-derived reﬂectance during November 2007. Intense smoke plumes were observed in the IRS-P4 OCM satellite data over IGP associated with agricultural crop residue burning during the study period. Terra-MODIS AOD and CALIPSO LIDAR backscatter datasets were analysed over the region to understand the spatial and temporal variation of the aerosol properties. Ground-based measurements on aerosol optical properties and black carbon (BC) mass concentration were carried out during 7–14 November 2007 over urban region of Hyderabad, India. Top of the Atmosphere (TOA) reﬂectance estimated from IRS-P4 Ocean Color Monitor (OCM) data showed large variations due to anthropogenic activities associated with crop residue burning and ﬁreworks. Atmospheric corrections to OCM satellite data using 6S radiative transfer code with inputs from ground and satellite measurements could account for the variations due to diﬀerential aerosol loading. Results are discussed in this paper. Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Agriculture residue burning; Diwali ﬁreworks; 6S; Aerosol; Smoke; LIDAR

1. Introduction Atmospheric aerosols play a potentially important role in the Earth’s climate system because of their direct interaction with solar and terrestrial radiation through scattering and absorption thereby modifying the Earth’s radiation budget. The knowledge of spatial and temporal distribution of aerosols on a global scale is important to understand the dynamics of aerosols and the associated inﬂuence on global climatic conditions (Tripathi et al., 2005). Globally, biomass burning is recognized as an important source of atmospheric pollution giving rise to the release of large quantities of gaseous emissions and particulate matter (Crutzen and Andreae, 1990). Each year more then * Corresponding author. Tel.: +91 40 23884220x4558; fax: +91 40 23879869. E-mail addresses: [email protected] (K.V.S. Badarinath), [email protected] (A.R. Sharma).

100 million tons of smoke aerosols are released into the atmosphere from biomass burning out of which 80% is in the tropical regions (Kharol and Badarinath, 2006). In India, forest ﬁre biomass constitutes only 8% of total biomass consumption and contributes to 16% of BC and 33% organic carbon (OC) emissions (Reddy and Venkataraman, 2002). Earlier studies over Amazonian region have demonstrated that during the dry season, vast forest ﬁres release the huge amounts of smoke, and consequently pyrogenic aerosol particles dominate the atmospheric aerosols over most of the Amazon Basin (Potter et al., 1998; Freitas et al., 2005) which causes pronounced seasonal changes over the region (Artaxo et al., 2002). In the Mediterranean basin, ﬁre activity regularly aﬀects several hundred thousand hectares with consequences on ecosystems and local air pollution (Lelieveld et al., 2002). Globally these ﬁres release large amounts of greenhouse gases (Liu et al., 2005) and contribute to the reduction of world biodiversity through the extinction of species populations. Furthermore, anthropogenic emissions from forest ﬁres

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have a negative impact on human health, as they contribute to respiratory and cardio-pulmonary diseases and mortality (Kharol and Badarinath, 2006). In India, each year agricultural crop residues are burnt in IGP during October/November (Badarinath et al., 2006). The Indo-Gangetic Plains are one of the world’s largest and highly agriculturally productive areas encompassing a vast area from 21.75°N, 74.25°E to 31.0°N, 91.50°E. The IGP is composed of the Indus (areas in Pakistan, and parts of Punjab and Haryana in India) and the Gangetic Plains (Uttar Pradesh (UP), Bihar, and West Bengal in India, Nepal and Bangladesh). It accounts for 21% of the land area of Indian subcontinent and holds nearly 40% of the total population. In the IGP region nearly 12 million hectares account for rice–wheat crop rotation. Harvesting of these crops with combine harvesters is very popular among the farmers of Punjab, Haryana and Western Uttar Pradesh. Crop residues after harvesting get burnt in these areas because of the high labour wages and anxiety of the farmers to get the crops produce collected and marketed as early as possible. Residues burning in the rice–wheat-rotation procedure has resulted in pollutant emission, loss of nutrients, diminished soil biota, and reduced total N and C in the topsoil layer (Kushwaha and Singh, 2005). Diwali is a major Indian festival, known as the ‘‘Festival of Lights,” it symbolizes the victory of good over evil, and lamps are lit as a sign of celebration and hope for humankind. It usually occurs in October/November, and is one of the most popular and eagerly awaited festivals in India. On the occasion of this festival, people burn crackers and sparkles to express their happiness which is the most unusual source of aerosol and trace guesses. High concentrations of anthropogenic aerosols and metals are injected into the atmosphere due to burning of crackers and ﬁreworks especially in urban regions. Fireworks mainly contain chemicals such as arsenic, sulfur, manganese, sodium oxalate, aluminium and iron dust powder, potassium perchlorate, strontium nitrate, barium nitrate and charcoal (Kulshrestha et al., 2004; Wang et al., 2007). Burning of ﬁreworks releases pollutants, like sulfur dioxide (SO2), carbon dioxide (CO2), carbon monoxide (CO), suspended particles (including particles below 10 lm in diameter, i.e. PM10) and several metals like aluminium, manganese and cadmium, etc., which are associated with serious health hazards. Due to the combined eﬀect of Diwali ﬁreworks and agricultural crop residue burning the entire belt of the Indo-Gangetic Plain was under inﬂuence of high aerosol loading during the study period. The atmospheric eﬀects due to anthropogenic perturbations contribute signiﬁcantly to the signal received by a multi-spectral satellite sensor (Hadjimitsis et al., 2004). The atmospheric constituents aﬀect the visible and nearinfrared satellite signals modifying the spectral and spatial distribution of the radiation incident on the surface. The Second Simulation of Satellite Signal in Solar Spectrum (6S) radiative transfer model is widely used for atmospheric

corrections to satellite data (Tachiri, 2005; Vermote and Saleous, 2006). It simulates the reﬂection of solar radiation by a coupled atmosphere–surface system for a wide range of atmospheric, spectral, and geometrical conditions. The parameters required to run the 6S radiative transfer model include AOD, water vapour and trace gases, etc. In the present study, the impact of anthropogenic aerosols on the satellite-derived reﬂectance over the Hyderabad region and its surroundings are studied using long-term satellite data from IRS-P4 Ocean Color Monitor (OCM) sensor, ground-based measurements and satellite-derived aerosol parameters. The main objective of the study is to estimate and correct for the eﬀect of anthropogenic emissions on satellite-derived reﬂectance. Atmospheric corrections to satellite data were implemented using radiative transfer model with inputs on MODIS-derived AOD/water vapour/OMI-Ozone which were validated with ground observations. 2. Data sets and methodology 2.1. Ground-based measurements AOD, water vapour and ozone measurement were carried out using a handheld multi-channel microprocessorcontrolled total ozone portable sun photometer (MICROTOPS-II) instrument in the premises of the National Remote Sensing Agency (NRSA) campus located at Balanagar (17o.280 N and 78o.260 E), Hyderabad. The sun photometer performs measurements at six spectral bands centered on 380, 440, 500, 675, 870 and 1020 nm (Morys et al., 2001). The sun photometer works on the principle of measuring the surface solar radiation intensity in the speciﬁed bands and calculates the corresponding optical depths by knowing the respective intensities at TOA. The details about the design, calibration, and performance of MICROTOPS-II have been described elsewhere (Morys et al., 2001; Badarinath et al., 2007a). Nighttime measurements of aerosol vertical proﬁle were carried out using the portable micro pulsed LIDAR operating at 532 nm. The LIDAR laser output pulse energy was set at 10 lJ with 2500 Hz pulse-repetition rate (Klett, 1981). The laser system bin width was set at 200 ns, corresponding to a height resolution of 30 m. The raw data proﬁle was integrated every 2-min interval. The design speciﬁcations of the portable LIDAR system are described elsewhere (Kumar, 2006). Continuous measurements of black carbon mass concentration (BC) were also carried out using an aethalometer; model AE-21 of Magee Scientiﬁc. The instrument aspirates ambient air at the height of 3 m above the ground with the aid of a pump. The BC mass concentration is estimated by measuring the change in the transmittance of a quartz-ﬁlter tape, on to which the particles impinge. The instrument had been operating at a time base of 5 min, continuously on the experiment days with a ﬂow rate of 3 l min1. The instrument has been factory-cali-

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brated and errors in the measurements are ±2% (Badarinath et al., 2007b). 2.2. Satellite data 2.2.1. MODIS The Moderate Resolution Imaging Spectroradiometer (MODIS) instrument on board the Earth Observing System (EOS) Terra/Aqua satellites (King et al., 1999) provides invaluable information about ﬁres and aerosol parameters over land. MODIS observes the earth in 36 spectral bands ranging from 0.4 to 14.4 lm with spatial resolutions of 250 m, 500 m and 1 km. The thermal-infrared sensing capability of MODIS sensor gives it the ability to detect active ﬁres with high temporal resolution. The detailed information on MODIS-derived ﬁre algorithm have been described elsewhere (Giglio et al., 2003). MODIS-derived ﬁre locations were downloaded over Indian region from (http://maps.geog.umd.edu/activeﬁre_html). MODIS-derived aerosol properties over land (Kaufman et al., 1997a; Chu et al., 1998, 2002; Ichoku et al., 2002) and over the ocean (Tanr´e et al., 1997; Remer et al., 2002) have been validated by more than 30 Aerosol Robotic Network (AERONET) stations worldwide. The detailed methodology of the retrieval of AOD has been discussed by Kaufman et al. (1997a). In the present study, Terra-MODIS Collection Version 005 AOD data (MOD08_D3.005) available from Giovanni-MODIS Online Visualization and Analysis System (MOVAS) site (http://g0dup05u.ecs.nasa.gov/Giovanni) were used to analyze spatial variations of the aerosol loading over the region during study period. 2.2.2. IRS-P4 (Oceansat-1) The OCM is a remote sensing payload on board the polar orbiting sun-synchronous Indian Remote Sensing Satellite IRS-P4, which was launched on 26 May 1999. OCM makes observations at eight narrow wavelength bands in the visible and near-IR wavelengths ranging from 400 to 885 nm (band 1: 402–422; band 2: 433–453; band 3: 480–500; band 4: 500–520; band 5: 545–565; band 6: 660– 680; band 7: 745–785; band 8: 845–885 nm); the radiometric resolution of the sensor is 12 bits. The spatial resolution of the pixel is 360 250 m at nadir and the swath is 1420 km. 2.2.3. CALIPSO The Cloud–Aerosol Lidar and Infrared Pathﬁnder Satellite Observation (CALIPSO) mission provides global proﬁling measurements of cloud and aerosol distribution and properties to improve understanding of weather and climate (Winker et al., 2003). CALIPSO was launched on 28 April 2006 as a part of A-Train satellite series. The CALIPSO payload consists of three instruments: the Cloud–Aerosol Lidar with Orthogonal Polarization (CALIOP), an Imaging Infrared Radiometer (IIR), and a moderate spatial resolution Wide Field-of-view Camera

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(WFC). CALIOP is a two-wavelength polarization-sensitive LIDAR which provides proﬁles of aerosols and clouds backscatter at 532 and 1064 nm, with 30-m vertical resolution and 333-m horizontal resolution. A detailed discussion on CALIOP data products has been described elsewhere (Vaughan et al., 2004). Individual band digital values (DN) from each band were converted into spectral radiance (Li) using sensor’s pre launch calibration coeﬃcients. The top of atmosphere (TOA) reﬂectance q(k)i for each spectral band was computed using the relation, qðkÞi ¼ pLi d 2 =E0 cos h

ð1Þ 2

where, Li is spectral radiance, d is earth–sun distance, E0 is the extra-terrestrial solar irradiance and h is the solar zenith angle. Atmospheric correction to the satellite data was performed using the Second Simulation of Satellite Signal in Solar Spectrum (6S) code (Vermote et al., 1997b). The 6S code predicts the reﬂectance of the objects at the TOA using information for the surface reﬂectance and atmospheric conditions. The parameters required to be deﬁned before running 6S code are water vapour, AOD, ozone, aerosol model, date and time of image acquisition, target elevation, solar zenith and azimuth angle, satellite view and azimuth angle, type of sensor and target elevation. Solar zenith and azimuth angle, satellite zenith and azimuth angle and date and time of the image acquisition were extracted from header ﬁle of OCM satellite data. Atmospheric parameters viz. aerosol optical depth, water vapour and ozone were collected using ground and satellite-based measurements over urban region of Hyderabad. The surface reﬂectance free from atmospheric eﬀects is computed according to (Mahiny and Turner, 2007). Ref ¼ ½ðAp þ B=½1 þ ðgðAp þ BÞÞ

ð2Þ

where A = 1/ab, B = q/b and p = TOA reﬂectance, a is the global gas transmittance, b is the total scattering transmittance, g is the spherical albedo and q is the atmospheric reﬂectance. 3. Results and discussion The present study deals with the impact of extensive anthropogenic activities associated with agricultural crop residues burning as well as a major Indian festivity known as ‘‘Diwali” (celebrated on 9 November 2007) on IRS-P4 (Oceansat-1) satellite-derived reﬂectance. Fig. 1 shows the MODIS-derived active ﬁre locations over the Indian region for the period 7–14 November 2007. It is clear from the ﬁgure that majority of ﬁres occurred in the western Indo-Gangetic Plain, mainly in Punjab state with a smaller amount in central India and negligible burning in the north-east and southern region of the country. These ﬁres are mainly attributed to the agricultural crop residue burning practices associated with rice–wheat system (Badarinath et al., 2006).

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Fig. 1. Images of active ﬁres from MODIS during 7–14 November 2007.

Fig. 2 shows the day average AOD500 variations with error bars showing the standard deviation of the daily averaged value, over the urban region of Hyderabad during the period 7–14 November 2007. It is clear from the ﬁgure, that the AOD500 values dramatically increased from 7 to 13 November 2007. The mean AOD showed smallest value

on 14 November 2007 indicating that the atmosphere was relatively clean compared to other days of measurement period. The higher AOD values (>0.4) compared to those on normal day (14 November 2007) during the measurement period suggested an additional loading of aerosols over the region. The peak observed in AOD500 on 11

Fig. 2. Day average variation of MICROTOPS-II sun photometer measured AOD500 with standard deviation over Hyderabad during 7–14 November.

Fig. 3. Day average variation of black carbon aerosol mass concentration over the urban region of Hyderabad.

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Fig. 4. LIDAR backscatter proﬁle over Hyderabad on 9 November 2007 (Diwali festival).

November 2007 is associated with extensive ﬁres observed over IGP and central region of India on 10 November 2007. NOAA HYSPLIT air mass back trajectory showed northerly winds over measurement site is suggesting longrange transport of anthropogenic aerosols over the region. Earlier studies (Kharol and Badarinath, 2006) over the urban region of Hyderabad suggested an AOD increase of 20% on burning days compared to normal days. Fig. 3 shows the day average variations of BC aerosol mass concentration with error bars showing the standard deviation of the daily averaged value, for the period 1–15 November 2007. It is clear from the ﬁgure that the diurnal average value of BC was varying between 4.8 and 7.7 lg/m3 during 2–6 November 2007 corresponding to normal conditions. The drastic increase was observed in BC concentration after 5th November and reached maximum value of 27 lg/m3 on 7 November. This sudden increase in BC concentrations attributed additional loading due to the extensive ﬁreworks and long-range transport of aerosols from agriculture crop residue burning. LIDAR measurements on aerosol backscatter at 532 nm were performed on 9 November 2007 night (Diwali day) to understand the vertical mixing of anthropogenic aerosols. The LIDAR aerosol backscatter proﬁle at 532 nm shown in Fig. 4 suggested increase in boundary layer height associated with convective activity due to aerosol emissions from ﬁreworks in the urban area of Hyderabad on Diwali festival night. High aerosol backscatter shown in red color1 near the surface occurred due to presence of dense aerosol layer formed by burning of crackers attributed that emissions of crackers burning is contributing up to 1 km. These elevated aerosol layers reach a maximum height of

1 For interpretation of the references to color in the text, the reader is referred to the web version of this article.

1.8 km around 19:00 h IST, which corresponds to peak time of ﬁreworks in the city and remained at an altitude of 1.2 km altitude till 10th November early morning. In order to identify the possible sources of high aerosol loading during the measurement period satellite datasets of CALIPSO (Cloud–Aerosol Lidar and Infrared Pathﬁnder Satellite) and MODIS AOD were analysed over the region. Fig. 5 shows the spatial distribution of TERRA-MODISderived AOD at 550 nm for the period 7–14 November 2007 over the Indian subcontinent. During the experiment period persistence of high AOD over the IGP and central India and lower AOD at north-east and southern region of the country were observed. This high aerosol loading in upper part of Indian subcontinent is mainly attributed to agriculture crop residues burning in the Indo-Gangetic Plain (shown in Fig. 1) and ﬁre works associated with the festive activities of Diwali. A good correlation between ﬁre counts and MODIS AOD550 over parts of Indian region were reported earlier (Badarinath et al., 2007c). Fig. 6 shows the available CALIPSO backscatter proﬁle covering the parts of Indo-Gangetic Plain (IGP) on 7 November 2007. The CALIPSO backscatter signals reveals strong aerosol backscattering from an altitude of 3 km covering an area from 18.90°N to 25°N Latitude and 85.46°E to 86.90°E longitude, this is indicative of higher aerosol concentrations due to anthropogenic activities. Fig. 7 shows IRS-P4 OCM false color composite (FCC) in 6:3:2 band combinations on 7 November, 9 November, 11 November and 13 November 2007, respectively, over the India. Intense smoke plume emanating from agricultural crop residues burning in the IGP area and hazy atmospheric conditions due to ﬁreworks over India are clearly visible. To investigate the impact of anthropogenic aerosols on satellite-derived reﬂectance, OCM retrieved TOA reﬂectance and AOD over the urban region of Hyderabad were

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Fig. 5. Spatial distribution of TERRA-MODIS-derived AOD at 550 nm for the period 7–14 November 2007 over the Indian subcontinent.

Fig. 6. CALIPSO derived backscatter image showing the aerosol loading over parts of Indo-Gangetic Plain (IGP) on 7 November 2007.

compared. Fig. 8a and c shows the TOA reﬂectance and AOD over the urban region of Hyderabad in IRS-P4 OCM band 1 (402–422 nm) and band 8 (845–885 nm). Over the urban region of Hyderabad, the TOA reﬂectance in OCM band 1 (Fig. 8a) showed increased values on 9, 11 and 13 November 2007 and lower values on 7, 8, 10, 12 and 14 November 2007. The higher TOA reﬂectance values are associated with high aerosol loading, whereas the lower

TOA reﬂectance values with comparatively low aerosol loading. Fig. 8c shows TOA reﬂectance in IRS-P4 OCM band 8 (845–885 nm) and AOD over Hyderabad. The TOA reﬂectance in NIR shows less variation compared to that in the visible spectrum suggesting that the aerosol path radiance has serious eﬀects on shorter wavelength bands (Paolini et al., 2006). This happens because the inﬂuence of aerosols and air molecules is smaller at the longer

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Fig. 7. IRS-P4 OCM FCC showing smoke plume (in circle) over Indo-Gangetic Plain (IGP) on 7, 9, 11 and 13 November 2007.

Fig. 8. TOA reﬂectance and aerosol optical depth variation over urban region of Hyderabad in (a) IRS-P4 OCM band 1 and (c) IRS-P4 OCM band 8 and scatter plot between aerosol optical depth and IRS-P4 OCM band 1 (b) and IRS-P4 OCM band 8 (d).

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Fig. 9. Comparison of TOA reﬂectance and atmospherically corrected reﬂectance over (a) water body of Nizam Sagar, (b) Hyderabad and (c) dense forest canopy in November 2007.

wavelengths. However, the atmospheric absorption due to water vapour, carbon dioxide, methane and other gases can play a signiﬁcant role in NIR region (Vermote and Saleous, 2006). Fig. 8b and d shows scatter plots between MICROTOPS AOD and TOA reﬂectance in IRS-P4 OCM band 1 and band 8. The TOA reﬂectance shows positive correlation with AOD conﬁrming the observation that the variation in TOA reﬂectance is due to the aerosol path radiance. The atmospheric aerosols contribute to the additional path radiance due to scattering/absorption and thus inﬂuence the TOA reﬂectance in all visible and near-IR spectral bands. Shorter wavelength visible bands are relative more inﬂuenced by atmospheric eﬀects compared to longer wavelengths and the availability of MODIS AOD data at 440, 550, 660 nm provides a means to carry out the corrections. To investigate the impact of anthropogenic aerosols on satellite-derived reﬂectance, various pseudo invariant features (PIFs) observation sites were selected and marked on IRS-P4 OCM FCC. The diﬀerent PIFs selected are (1) water body of Nizam Sagar, (2) dense forest canopy in the surroundings of Hyderabad and (3) urban area of Hyderabad. Fig. 9 shows a comparison of TOA reﬂectance and atmospherically corrected reﬂectance over (a) water body of Nizam Sagar, (b) the urban area of Hyderabad and (c) a dense forest canopy. The variations in surface reﬂectance due to increased atmospheric loading of pollutants associated with biomass burning and other festivities

could be clearly seen in Fig. 9. The satellite data sets corrected for the atmospheric eﬀects provides actual variations of the land cover feature under consideration on diﬀerent days. Atmospheric correction removes the artifacts and is useful in studies related to crop growth, drought monitoring using NDVI, etc. It is clear from the Figure that the TOA reﬂectance during the measurement period showed large variation over all three pseudo invariant features (PIFs) observation sites. The atmospheric correction applied to satellite data using the 6S model accounted for the variations due to variable aerosol loading. The satellite data corrected for atmospheric eﬀects show much less variations in reﬂectance over the PIFs on diﬀerent days during the measurement period. The reﬂectance values after the atmospheric correction suggests true representation on the ground suggesting the importance of atmospheric corrections over the tropics with variable aerosol loading due to anthropogenic activities.

4. Conclusions The present study has been conducted to analyze the impact of extensive anthropogenic activity associated with Indian Diwali festival and agricultural crop residues burning in IGP on aerosol characteristics and satellite-derived reﬂectance over India during 7–14 November 2007 using

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satellite data and ground-based measurements. Results of the study suggest that: The observed AOD values were high during the biomass burning and festive period. The BC concentration levels increased drastically and the levels remained high for approximately 6 days after festivities. Nighttime micro pulse LIDAR observations showed elevated layers at a height of 1.2 km on 9 November 2007 due to anthropogenic emissions associated with ﬁre works and burning of crackers. Enhanced aerosol emissions from anthropogenic activities aﬀect the satellite-derived reﬂectance signiﬁcantly and atmospheric corrections to satellite data could account for the path radiance due to elevated aerosol loading. Acknowledgements The authors thank Director, NRSC and DD, RS & GIS-AA for their necessary help and encouragement. The authors thank ISRO chairman for their necessary support. References Artaxo, P., Martins, J.V., Yamasoe, M.A., Proco´pio, A.S., Pauliquevis, T.M., Andreae, M.O., Guyon, P., Gatti, L.V., Leal, A.M.C. Physical and chemical properties of aerosols in the wet and dry season in Rondonia, Amazonia. J. Geophys. Res. 107 (D20), 8081, doi:10.1029/ 2001JD000666, 2002. Badarinath, K.V.S., Kharol, S.K., Kaskaoutis, D.G., Kambezidis, H.D. Inﬂuence of atmospheric aerosols on solar spectral irradiance in an urban area. J. Atmos. Solar Terr. Phys. 69, 589–599, 2007a. Badarinath, K.V.S., Kharol, S.K., Kiran Chand, T.R., Parvathi, Y.G., Anasuya, T., Jyothsna, A.N. Variations in black carbon aerosol, carbon monoxide and ozone over an urban area of Hyderabad, India during the forest ﬁre season. Atmos. Res. 85, 18–26, 2007b. Badarinath, K.V.S., Kharol, S.K., Kiran Chand, T.R. Use of satellite data to study the impact of forest ﬁres over the northeast region of India. IEEE Geosci. Remote Sens. Lett. 4 (3), 485–489, 2007c. Badarinath, K.V.S., Kiran Chand, T.R., Prasad, V.K. Agriculture crop residue burning in the Indo-Gangetic plains – a study using IRS P6 AWiFS satellite. Curr. Sci. 91, 1085–1089, 2006. Chu, D.A., Kaufman, Y.J., Remer, L.A., Holben, B.N. Remote sensing of smoke from MODIS airborne simulator during the SCAR-B experiment. J. Geophys. Res. 103, 31979–31987, 1998. Chu, D.A., Kaufman, Y.J., Ichoku, C., Remer, L.A., Tanr´e, D., Holben, B.N. Validation of MODIS aerosol optical depth retrieval over land: ﬁrst result and evaluation of aerosol from the Terra Spacecraft (MODIS). Geophys. Res. Lett. 29 (12), 1–4, 2002. Crutzen, P.J., Andreae, M.O. Biomass burning in the tropics: impact on atmospheric chemistry and biogeochemical cycles. Science 250, 1669– 1678, 1990. Freitas, S.R., Longo, K.M., Silvadias, M.A.F., Silvadias, P.L., Chatﬁeld, R., Prins, E., Artaxo, P., Grell, G.A., Recuero, F.S. Monitoring the transport of biomass burning in South America. Environ. Fluid Mech. 5, 135–167, 2005. Giglio, L., Descloitres, J., Justice, C.O., Kaufman, Y.J. An enhanced contextual ﬁre detection algorithm for MODIS. Remote Sens. Environ. 87, 273–282, 2003.

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Advances in Space Research 43 (2009) 1555–1562 www.elsevier.com/locate/asr

Variation of ionospheric total electron content in Indian low latitude region of the equatorial anomaly during May 2007–April 2008 Sanjay Kumar, A.K. Singh * Atmospheric Research Lab., Department of Physics, Banaras Hindu University, Lanka, Varanasi 221005, India Received 12 September 2008; received in revised form 27 January 2009; accepted 28 January 2009

Abstract The ionospheric total electron content (TEC), derived by analyzing dual frequency signals from the Global Positioning System (GPS) recorded near the Indian equatorial anomaly region, Varanasi (geomagnetic latitude 14°, 550 N, geomagnetic longitude 154°E) is studied. Speciﬁcally, we studied monthly, seasonal and annual variations as well as solar and geomagnetic eﬀects on the equatorial ionospheric anomaly (EIA) during the solar minimum period from May 2007 to April 2008. It is found that the daily maximum TEC near equatorial anomaly crest yield their maximum values during the equinox months and their minimum values during the summer. Using monthly averaged peak magnitude of TEC, a clear semiannual variation is seen with two maxima occurring in both spring and autumn. Statistical studies indicate that the variation of EIA crest in TEC is poorly correlated with Dst-index (r = 0.03) but correlated well with Kp-index (r = 0.82). The EIA crest in TEC is found to be more developed around 12:30 LT. Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Total electron contents (TECs); Equatorial ionization anomaly (EIA); Global Positioning System (GPS)

1. Introduction The changes in the temporal and spatial features of total electron contents (TECs) at the equatorial and low latitude regions are signiﬁcant compared to relatively small at mid latitude regions (Davies, 1980). The equatorial and low latitude ionospheres are very dynamic due to the various processes associated with the phenomenon of equatorial ionization anomaly (EIA) and equatorial spread-F (ESF) irregularities in these regions. The equatorial ionospheric anomaly (EIA) is characterized, in terms of latitudinal distribution of ionization, by a trough at the magnetic equator and crests at about ±17° magnetic latitude (Appleton, 1946); the crest-to-trough ratio (about 1.5 in the day time peak electron density) and the position of the crests vary with various geophysical conditions. Many theories, like the diﬀusion theory (Mitra, 1946) and the electrodynamic drift theory (Martyn, 1955; *

Corresponding author. Tel.: +91 542 2313431. E-mail address: abhay_s@rediﬀmail.com (A.K. Singh).

Duncan, 1960), have been put forward to explain the anomaly. The diﬀusion theory has been shown to be important but not suﬃcient to explain the observations (Rishbeth et al., 1963). The electrodynamic drift theory, on the other hand, has been successful in explaining the observations (Bramley and Peart, 1965; Moﬀett and Hanson, 1965). According to the drift theory, the north–south geomagnetic ﬁeld combined with the daytime east–west ionospheric electric ﬁeld (both being parallel to the Earth’s surface at the equator) generates a plasma fountain which rises to several hundred kilometers at the equator. When the upward drifting plasma loses its momentum, it diﬀuses under gravity along the geomagnetic ﬁeld lines to higher latitudes forming the crests (Hanson and Moﬀett, 1966). The fountain and resulting anomaly can be cover more than 30° latitudes on either side of the magnetic equator (Balan and Bailey, 1995). Balan and Bailey (1995) further studied the plasma fountain including neutral wind also and shown that the plasma velocity turning more poleward in that hemisphere, where the wind is poleward. The formation of the EIA is seen in the total electron content (TEC), which is the integral of electron number

0273-1177/$36.00 Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2009.01.037

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density along the line of sight from satellite to receiver and can vary dramatically from day-to-day (Huang et al., 1989; Rastogi and Klobuchar, 1990). The latitudinal variation of daily TEC was found to correlate with strength of the electrojet current (Rastogi and Klobuchar, 1990). On the other hand long-term averages of TEC do have seasonal variations. The strength of monthly anomaly crest increases with solar activity and shows a winter anomaly with the winter strength larger than the summer strength for all solar activity levels (Huang and Cheng, 1996). The strength of equatorial anomaly shows semiannual variations. In the Indian sector, the diurnal variation in TEC was reported by Rama Rao et al. (2006) who showed that at EIA crest, day maximum in TEC occurs between 13:00 and 16:00 LT and a short lived day minimum occurs between 05:00 and 06:00 LT. He further showed that in the Indian sector the EIA crest is found to occur in the latitude zone of 15° to 25°N geographic latitudes (5° to 15°N geomagnetic latitudes). The seasonal and geomagnetic eﬀects on the equatorial ionospheric anomaly crest by analyzing TEC data acquired from a chain of nine ground GPS stations at and in the neighborhood of Taiwan during the solar minimum period between September 1996 and August 1997, was studied by Wu et al. (2004). They found that the surveyed data indicates semiannual variation in TEC at anomaly, with maxima at equinoxes. They further demonstrated that the monthly EIA was well correlated with Dst and weakly correlated with F10.7. Huang and Cheng (1996) studied solar cycle variations of the equatorial anomaly in TEC received by single ground station of Lumping (25.00°N, 12.17°E). They found that the winter crest appears larger and earlier than the summer crest and the summer crest appears lower in latitude than during other seasons. They attributed these seasonal eﬀects to the daytime meridional wind. Apart from several individual measurements of TEC from various locations in India (Rastogi and Sharma, 1971; DasGupta and Basu, 1973; Rastogi et al., 1975; Rama Rao et al., 1977; Davies et al., 1979), there were no coordinated, continuous measurements of TEC from all the latitude zones of the Indian region, except the recent GAGAN project (Rama Rao et al., 2006). It was observed that there were very few studies of ionospheric TEC measurements at low latitudes Indian regions. In this paper, we are trying to study seasonal and day-to-day variability of equatorial ionospheric anomaly (EIA) in TEC by GPS measurement at Indian low latitude station Varanasi (geomagnetic latitude 14°, 550 N, longitude 154°E) for complete one year from May 2007 to April 2008. To study the eﬀect of geomagnetic activity, we have taken Dst-index and Kp-index and compared our EIA crest variations. The eﬀect of solar activity was also studied considering sunspot number (SSN) data and solar ﬂux F10.7 data. We described experimental observation of TEC measurement using GPS in Section 2. Our main results are given in Section 3 which describes the seasonal variation as well as eﬀect of solar and magnetic activity. The discussions of these results are presented in Section 4. The last Section 5 summarizes the results.

2. Experimental observation The GPS navigation system comprises of three distinct ‘segments’ (Sonnenberg, 1988; Ackrroyd and Lorimer, 1990). The Global Positioning System consists of 24 satellites, called ‘space segment’, distributed in six orbital planes, around the globe at an altitude of 20,200 km and orbital period of 12 h. Each satellite transmits signals on two frequencies (f1 = 1575.42 MHz and f2 = 1227.60 MHz) with two diﬀerent codes P1 (or C/A) and P2 and two diﬀerent carrier phases, L1 and L2 both being derived from a 10.23 MHz common oscillator. The second is the ‘control segment’, which includes ground stations, used for monitoring satellites and sending signals upward for the engineering control of each satellite and its transmitted codes and waveforms. The third segment is the ‘user segment’, which includes a GPS receiver. The GPS receiver calculates its position by selecting optimum conﬁguration of four satellites and by ﬁnding its range with respect to each satellite. The range is determined from the delay in the time taken for the signals to travel from each of satellites to the receiver, as measured by the diﬀerence between satellite transit time, which is known, and the signal reception time which is measured by autocorrelation. However, this diﬀerence does not take into account any error in the receiver’s clock relative to satellite’s clock and therefore the range is only approximate and is therefore called a pseudorange (Rama Rao et al., 2006). The ionosphere has a refractive index at radio frequencies, which is diﬀerent from unity and can aﬀect GPS signals in a number of ways as they pass from satellite to ground receiver (Coco, 1991; Wanninger, 1993; Klobuchar, 1996). One of the signiﬁcant eﬀects is that the GPS signals traversing the ionosphere undergoes an additional delay proportional to the total electron content (TEC), which is deﬁned as total number of free electrons in column of 1 m2 cross-sectional area along the ray path from the satellite to receiver. The GPS data provides an eﬃcient way to estimate TEC values with greater spatial and temporal coverage (Davies and Hartmann, 1997; Hocke and Pavelyev, 2001). Since the frequencies that are used in the GPS system are suﬃciently high, the signals are minimally aﬀected by the ionospheric absorption and Earth’s magnetic ﬁeld, both in the short-term, as well as in the long-term changes in the ionospheric structure. In the present study, the slant total electron content (STEC) data derived from GPS data recorded at our low latitude station Varanasi is converted into vertical total electron content (VTEC) according to (Rama Rao et al., 2006). VTEC ¼ ðSTEC ½bR þ bS Þ=SðEl Þ

ð1Þ

where bR and bS are receiver and satellite biases, respectively, El is the elevation angle of the satellite in degrees, S(El) is the obliquity factor with zenith angle v at the ionospheric pierce point (IPP) and VTEC is the vertical TEC at the IPP. The obliquity factor S(El) (or mapping func-

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tion) is deﬁned as (Mannucci et al., 1993; Langley et al., 2002). ( 2 )1=2 1 RE CosðEl Þ ð2Þ ¼ 1 SðEl Þ ¼ CosðvÞ RE þ h where RE is the mean Earth’s radius in km, h is the ionospheric eﬀective height above the Earth’s surface, v is the zenith angle and El is the elevation angle of satellite in degree. The errors that aﬀect the STEC measurements, which is then propagated to the VTEC, is directly associated to the

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errors in the computation of the receiver and satellite biases. The biases for the satellites and the receiver used in the present work were computed using the La Plata Ionospheric Model (Brunini et al., 2008). According to Ciraolo et al. (2007), the mean error on the STEC should be in the order of ±3 TECU. The VTECs derived from a GPS receiver (Trimble 5700) installed at Varanasi, during the period, May 2007 to April 2008 are used to quantify the equatorial anomaly in the present paper. The calculated VTEC for all PRN for a typical day 1 March 2008 is shown in Fig. 1(a). The location of

Fig. 1. (a) The diurnal variation of VTEC for a typical day (1 March 2008). (b) Path of GPS satellite for a typical day (1 March 2008) passing over Varanasi, a low latitude station near the EIA crest.

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Varanasi and path of GPS satellites, for a typical day 1 March 2008 is shown in the Fig. 1(b). From this ﬁgure we observe that the peak in VTEC (47.5 TECU) occurs around 13:46 LT (IST). To study the diurnal, seasonal and annual variation of EIA, and the eﬀect of geomagnetic activity and solar activity on their variation, we have chosen daily peak value of TEC observed at Varanasi (a station situated near EIA crest, in Indian region) as the daily anomaly crest value. The value of daily anomaly crest was identiﬁed for each of 365 days as a function of local time. To see the eﬀect of geomagnetic activity on the variation of anomaly crest value, we have taken hourly Dst-index (http://swdcwww. kugi.kyto-u.ac.jp) and then taken the average of each day and hence averaged for each month. Three hourly Kpindex data have also been used (http://swdcwww.kugi. kyto-u.ac.jp) and average is taken in similar manner as done for Dst-index. To study the eﬀect of solar activity, sunspot number (SSN) and solar ﬂux F10.7 data have been taken (http://www.ngdc.noaa.gov) for period May 2007 to April 2008. 3. Results To study diurnal and seasonal variation of EIA crest in TEC we have plotted the daily peak value of TEC as EIA crest, which is shown in Fig. 2. It is clear from Fig. 2 that

variation in EIA crest in TEC shows semiannual variation with two maxima in equinox months (October, April) and two minima in winter (December) and summer (July). To demonstrate seasonal variations of EIA crest in TEC in more details and to study the solar and geomagnetic activity, we have taken the monthly average of EIA crest in TEC, Dst-index, Kp-index, F10.7 and SSN and these monthly average values are plotted in Fig. 3. Here vertical bar indicates the standard deviation in the mean data. To see the eﬀect of solar and geomagnetic activity on the variation of anomaly crest the variation of monthly averaged peak TEC is compared with monthly average of solar index SSN, F10.7 and geomagnetic indices Dst and Kp, which is discussed in the following section. It is clear from Fig. 3 that variation in monthly average EIA crest in TEC is very similar to variation in monthly average Kp-index. The correlation coeﬃcient between monthly average EIA crest in TEC and monthly average Kp-index is found to be r = 0.82 and both shows semiannual variation. The monthly average Dst-index do not follow the variation as seen in monthly EIA crest in TEC (r = 0.03). The variation in monthly EIA crest in TEC is not much related with variation in solar indices F10.7 (r = 0.25) and SSN (r = 0.09) because our study period is a solar minimum period. Fig. 4 shows the correlation of EIA crest in TEC with Dst, Kp, SSN and F10.7. As expected from our previous

Fig. 2. Daily variation of EIA crest in TEC (peak VTEC), during the solar minimum period between May 2007 and April 2008.

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Fig. 3. Monthly mean variation of EIA crest in TEC, Dst-index, Kp-index, F10.7 and sunspot number (SSN) during the solar minimum period between May 2007 and April 2008.

results, monthly EIA crest in TEC is well correlated with Kp-index (r = 0.82) and both data indicates similar semiannual variations. The geomagnetic Dst-index showed poor correlation (r = 0.03) and there was no correlation between EIA crest in TEC with SSN and F10.7. Fig. 5 shows the occurrence of daily EIA crest in TEC in terms of local time. The daily EIA crest in TEC occurred over wide range of local time (10:00–18:00 LT). We ﬁt the histogram by a Gaussian function, a0 exp(z2/2) where z = (x a1)/a2 and x stands for local time (a0, a1, a2) = (51.994, 12.544, 0.064). The statistical mean location is at 12:30 LT and the full width at half maximum (FWHM) is 3.78 h in LT. Our all above reported results have been found in good agreement with the results reported by other authors in EIA crest regions (Huang and Cheng, 1996; Wu et al., 2004, 2008). 4. Discussions In the present study, we have analyzed the GPS derived TEC data to study the variation of daily EIA crest in TEC

observed at Varanasi during the solar minimum period of May 2007 to April 2008. The daily EIA crest in TEC shows both the seasonal and semiannual variation, which shows maxima in equinox (October and April) months and minimum in winter (December) and summer (July). Similar type of seasonal and annual variations in EIA crest have been studied by Huang and Cheng (1996), who also analyzed TEC data from a single receiver in Taiwan and found that the winter crest appears to be larger and earlier than the summer one and the latitudinal location of the crest was lowest in summer. Wu et al. (2004) also analyzed daily characteristics in vicinity of Taiwan during 22nd solar activity minimum from September 1996 to August 1997 and observed the similar results. Thus, the smaller anomaly crest in winter and summer is found in our work is consistent with their ﬁnding that the anomaly is least developed in winter and summer. In our study the sunspot number, SSN (r = 0.03) and solar ﬂux F10.7 (r = 0.25) have very little eﬀect on the variation of the EIA crest in TEC. This may be due to our analysis period of May 2007 to April 2008, which was a solar minimum period. Similar correlation of EIA crest in TEC with F10.7 (r = 0.38 for 1996 and

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Fig. 4. Correlation coeﬃcients comparing monthly mean EIA crest in TEC (peak VTEC) with monthly (a) Dst, (b) Kp, (c) SSN and (d) F10.7.

r = 0.04 for 1999) was reported by Wu et al. (2008) and r = 0.09 for 1996–97 by Wu et al. (2004) for Taiwan. It is well accepted that the fountain eﬀect is the main cause of the equatorial anomaly. Photoionization caused by solar EUV radiation also plays an important role in

the production of the anomaly. Photoionization can produce more electrons and therefore enhances the background electron density (Huang and Cheng, 1996; Wu et al., 2004). The semiannual variation in EIA crest in TEC is a combined eﬀect of the solar zenith angle and

Fig. 5. Histogram of daily EIA crest in TEC and local time.

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magnetic ﬁeld geometry. In general, the electron population in the ionosphere is mainly controlled by solar photoionization and recombination processes; whereas localized enhancements and depletions of electrons in the ionosphere are caused by electromagnetic forcing (Wu et al., 2004). During the equinox months, the subsolar point is around the equator, where the eastward electrojet-associated electric ﬁeld is often largest. Thus, in equinox months due to collocation of the peak photoelectron abundance and the most intense eastward electric ﬁeld regions, one would expect that the fountain eﬀect should be developed the most (Wu et al., 2004, 2008). On the other hand, during the solstices (winter and summer months), photoelectrons at the equator decreases because the subsolar points moves to higher latitudes and fountain eﬀect is expected to wane. The seasonal variations of the monthly averaged values of peak TEC can be explained based on a simple neutral atmosphere model (Bramley and Young, 1968), which explains that a meridional component of neutral wind blows from the summer to the winter hemisphere. During the summer solstice, the neutral wind can reduce the crest value by neutral drag as it blows in an opposite direction to the plasma diﬀusion process originating from the magnetic equator. This results in a low ionization crest value (Stening, 1992). At the equinoxes, the latitude for the midday overhead sun occurs in the region near the magnetic equator. The meridional wind blowing poleward should result in a high ionization crest value. Based on this scenario, a seasonal eﬀect on the crests should be expected, with the crest maximum at the equinoxes and minimum in the summer (Wu et al., 2004). Recently, Vijaya Lekshmi et al. (2008) have studied the relative eﬀects of the main drivers of the positive ionospheric storm (penetrating daytime eastward electric ﬁeld and direct and indirect eﬀects of equatorward neutral wind) using the Sheﬃeld University Plasmasphere Ionosphere Model (SUPIM). Their modeled results show that the penetrating daytime (morning–noon) eastward electric ﬁeld shifts the equatorial ionization anomaly crests in NmF2 and TEC to higher than normal latitudes and reduces their values at latitudes at and within the anomaly crests while the direct eﬀects of the equatorward wind (that reduce poleward plasma ﬂow and raise the ionosphere to high altitudes of reduced chemical loss) combined with the daytime production of ionization increase NmF2 and TEC at latitudes poleward of the equatorial region. The downwelling (indirect) eﬀect of the wind increases NmF2 and TEC at low latitudes while its upwelling (indirect) eﬀect reduces NmF2 and TEC at mid latitudes. The signiﬁcant result of this study is the good correlation of the monthly values of the EIA crest in TEC with the monthly Kp-index (r = 0.82). The correlation between the monthly values of the EIA crest in TEC and the monthly Dst-index is very much smaller (r = 0.03). Wu et al. (2008) have also observed good correlation of EIA crest in TEC with Kp-index in year 1994 (r = 0.84) and 1999 (r = 0.75) whereas for their complete study of 10 years

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(1994–2003) they found correlation with Kp-index as r = 0.29. Wu et al. (2004) for nine observational sites clustered around Taiwan found correlation of EIA crest in TEC with Kp-index as r = 0.41 for one year period from September 1996 to August 1997. Wu et al. (2008) further shown that a general good correlation of the monthly values of the anomaly crest with the monthly Dst-index in the period of low solar activity. However, correlation coeﬃcient is very bad for the long-term statistical study (r = 0.22). Whereas, Wu et al. (2004) found well correlation of EIA crest in TEC with Dst-index (r = 0.72) for only one year study. 5. Conclusion We have investigated seasonal, magnetic and solar eﬀects on characteristic of EIA by analyzing GPS derived TEC near the EIA crest station Varanasi. The following points emerge from our study: 1. The value of EIA crest in TEC shows seasonal variation with maximum occurrence in equinox months and minimum in winter and summer. It may be due to EEJ strength, which maximizes in equinox months and minimum in summer and winter. 2. The variation in EIA crest in TEC shows semiannual variation, which may be due to a combined eﬀect of the solar zenith angle and magnetic ﬁeld geometry. 3. The seasonal variation in EIA crest in TEC is likely inﬂuenced by seasonal variation of Kp-index and there is less dependence on Dst-index. 4. The variation of EIA crest in TEC is poorly correlated by solar activity indices (sunspot number and F10.7) because our study period was of solar minimum period. 5. The EIA crest is more developed around 12:30 LT. Our all above reported results have been found in good agreement with the results reported by other workers in the EIA regions. Acknowledgements This work is partly supported by Ministry of Earth Sciences, New Delhi under SERC project (MoES/P.O.(Seismo)/GPS/61/2006) and partly by UGC, New Delhi under Major Research Projects. References Ackrroyd, N., Lorimer, R. A GPS User’s Guide. Lloyd’s of London Press, 1990. Appleton, E.V. Two anomalies in the ionosphere. Nature 157, 691, 1946. Balan, N., Bailey, G.J. Equatorial plasma fountain and its eﬀects: possibility of an additional layer. J. Geophys. Res. 100, 21421– 21432, 1995. Bramley, E.N., Young, M. Winds and electromagnetic drifts in the equatorial F2-region. J. Atmos. Terr. Phys. 30, 99–111, 1968. Bramley, E.N., Peart, M. Eﬀect of ionization transport on the equatorial F region. Nature 206, 1245–1246, 1965.

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Brunini, C., Meza, A., Gende, M., Azpilicueta, F. South American regional maps of vertical TEC computed by GESA: a service for the ionospheric community. Adv. Space Res. 42, 737–744, 2008. Ciraolo, L., Azpilicueta, F., Brunini, C., Meza, A., Radicela, S.M. Calibration errors on experimental slant total electron content determined with GPS. J. Geodesy Springer 2 (81), 111–120, 2007. Coco, D. GPS satellites of opportunity for ionospheric monitoring. GPS World, 47, 1991. DasGupta, A., Basu, A. Investigation of ionospheric electron content in the equatorial region as obtained by beacon satellites. Ann. Geophys. 29, 409–419, 1973. Davies, K., Donnelly, R.F., Grubb, R.N., Rama Rao, P.V.S. ATS-6 satellite radio beacon measurements at ootacamund, India. Radio Sci. 14, 85–95, 1979. Davies, K., Hartmann, G.K. Studying the ionosphere with Global Positioning System. Radio Sci. 32, 1695–1703, 1997. Davies, K. Recent progress in satellite radio beacon studies with particular emphasis on the ATS-6 radio beacon experiment. Space Sci. Rev. 25, 357–430, 1980. Duncan, R.A. The equatorial F-region of the ionosphere. J. Atmos. Terr. Phys. 18, 89–100, 1960. Hanson, W.B., Moﬀett, R.J. Ionization transport eﬀects in the equatorial F region. J. Geophys. Res. 71, 5559–5572, 1966. Hocke, K., Pavelyev, A.G. General aspect of GPS data use for atmospheric science. Adv. Space Res. 27, 1313–1320, 2001. Huang, Y.N., Cheng, K., Chen, S.W. On the equatorial anomaly of the ionospheric total electron content near the northern anomaly crest region. J. Geophys. Res. 94, 13515, 1989. Huang, Y.N., Cheng, K. Solar cycle variation of equatorial ionospheric anomaly in total electron content in the Asian region. J. Geophys. Res. 101, 24513–24520, 1996. Klobuchar, J.A. Ionospheric eﬀects on GPS, in: Parkinson, B.W., Spilker, J.J. (Eds.), Global Positioning System: Theory and Applications, vol. 2. Progress in Astronautics and Aeronautics, Vol. 164, AIAA, Washington, p. 485, 1996. Langley, R., Fedrizzi, M., Paula, E., Santos, M., Komjathy, A. Mapping the low latitude ionosphere with GPS. GPS World 13 (2), 41–46, 2002. Mannucci, A.J., Wilson, B.D., Ewards, C.D. A new method for monitoring the Earth’s ionosphere total electron content using the GPS global network, in: Proceedings of ION GPS-93. Inst of Navigation, pp. 1323–1332, 1993.

Mitra, S.K. Geomagnetic control region F2 of the ionosphere. Nature 158, 668–669, 1946. Martyn, D.F. Theory of height and ionization density changes at the maximum of a chapman-like region, taking account of ion production, decay, diﬀusion and total drift, in: Proceedings Cambridge Conference, Physical Society, London, pp. 254–259, 1955. Moﬀett, R.J., Hanson, W.B. Eﬀect of ionization transport on the equatorial F region. Nature 206, 705–706, 1965. Rama Rao, P.V.S., Gopi Krishna, S., Niranjan, K., Prasad, D.S.V.V.D. Temporal and spatial variation in TEC using simultaneous measurements from the Indian GPS network of receivers during the low solar activity period of 2004–2005. Ann. Geophys. 24, 3279–3292, 2006. Rama Rao, P.V.S., Srirama Rao, M., Satyam, M. Diurnal and seasonal trends in TEC values observed at Waltair. Indian J. Radio Space Phys. 6, 233–235, 1977. Rastogi, R.G., Sharma, R.P. Ionospheric electron content at Ahmadabad (near the crest of equatorial anomaly) by using beacon satellite transmission during half a solar cycle. Planet. Space Sci. 19, 1505– 1517, 1971. Rastogi, R.G., Iyer, K.N., Bhattacharyya, J.C. Total electron content of the ionosphere over the magnetic equator. Curr. Sci. 44, 531–533, 1975. Rastogi, R.G., Klobuchar, J.A. Ionospheric electron content within the equatorial F2-layer anomaly belt. J. Geophys. Res. 95, 19045, 1990. Rishbeth, H., Lyon, A.J., Peart, M. Diﬀusion in the equatorial F-layer. J. Geophys. Res. 68, 2559–2569, 1963. Sonnenberg, S. Radio and Electronic Navigation, sixth ed Buterworth and Co., Chapter 7, 1988. Stening, R.J. Review paper modeling the low latitude F-region. J. Atmos. Terr. Phys. 54, 1387–1421, 1992. Vijaya Lekshmi, D., Balan, N., Vaidyan, V.K., Alleyne, H., Bailey, G.J. Response of the ionosphere to super geomagnetic storms: observations and modeling. Adv. Space Res. 41, 548–555, 2008. Wanninger, L. Eﬀects of the equatorial ionosphere on GPS. GPS World, 48, 1993. Wu, C.C., Fry, C.D., Liou, K., Tseng, C.L. Annual TEC variation in the equatorial anomaly region during the solar minimum: September 1996–August 1997. J. Atmos. Terr. Phys. 66, 199–207, 2004. Wu, C.C., Fry, C.D., Liou, K., Shan, Shao-Ju, Tseng, C.L. Variation of ionospheric total electron content in Taiwan region of the equatorial anomaly from 1994 to 2003. Adv. Space Res. 41, 611–616, 2008.

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Surface temperature estimation in Singhbhum Shear Zone of India using Landsat-7 ETM+ thermal infrared data P.K. Srivastava a, T.J. Majumdar b,*, Amit K. Bhattacharya a b

a Department of Geology & Geophysics, Indian Institute of Technology, Kharagpur 721 302, India Earth Sciences and Hydrology Division, Marine and Earth Sciences Group, Remote Sensing Applications Area, Space Applications Centre (ISRO), Ahmedabad, Gujarat 380 015, India

Received 20 June 2007; received in revised form 30 July 2008; accepted 21 January 2009

Abstract Land surface temperature (LST) is an important factor in global change studies, heat balance and as control for climate change. A comparative study of LST over parts of the Singhbhum Shear Zone in India was undertaken using various emissivity and temperature retrieval algorithms applied on visible and near infrared (VNIR), and thermal infrared (TIR) bands of high resolution Landsat-7 ETM+ imagery. LST results obtained from satellite data of October 26, 2001 and November 2, 2001 through various algorithms were validated with ground measurements collected during satellite overpass. In addition, LST products of MODIS and ASTER were compared with Landsat-7 ETM+ and ground truth data to explore the possibility of using multi-sensor approach in LST monitoring. An image-based dark object subtraction (DOS3) algorithm, which is yet to be tested for LST retrieval, was applied on VNIR bands to obtain atmospheric corrected surface reﬂectance images. Normalized diﬀerence vegetation index (NDVI) was estimated from VNIR reﬂectance image. Various surface emissivity retrieval algorithms based on NDVI and vegetation proportion were applied to ascertain emissivities of the various land cover categories in the study area in the spectral range of 10.4–12.5 lm. A minimum emissivity value of about 0.95 was observed over the reﬂective rock body with a maximum of about 0.99 over dense forest. A strong correlation was established between Landsat ETM+ reﬂectance band 3 and emissivity. Single channel based algorithms were adopted for surface radiance and brightness temperature. Finally, emissivity correction was applied on ‘brightness temperature’ to obtain LST. Estimated LST values obtained from various algorithms were compared with ﬁeld ground measurements for diﬀerent land cover categories. LST values obtained after using Valor’s emissivity and single channel equations were best correlated with ground truth temperature. Minimum LST is observed over dense forest as about 26 °C and maximum LST is observed over rock body of about 38 °C. The estimated LST showed that rock bodies, bare soils and built-up areas exhibit higher surface temperatures, while water bodies, agricultural croplands and dense vegetations have lower surface temperatures during the daytime. The accuracy of the estimated LST was within ±2 °C. LST comparison of ASTER and MODIS with Landsat has a maximum diﬀerence of 2 °C. Strong correlation was found between LST and spectral radiance of band 6 of Landsat-7 ETM+. Result corroborates the fact that surface temperatures over land use/land cover types are greatly inﬂuenced by the amount of vegetation present. Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: DOS; NDVI; LST; Emissivity; Vegetation proportion; Spectral radiance

1. Introduction Land surface temperature (LST), controlled by the surface energy balance, atmospheric state, thermal properties *

Corresponding author. Tel.: +91 79 26914304; fax: +91 79 2691 5825. E-mail address: [email protected] (T.J. Majumdar).

of the surface and subsurface, is an important parameter in many environmental models (Becker and Li, 1990), such as, energy and material exchange between atmosphere and land, weather forecasting, global ocean-ﬂow cycle and climate change. Thermal infrared (TIR) remote sensing is the only possible approach to retrieve LST (Coll et al., 2005) over large portions of the Earth surface at diﬀerent

0273-1177/$36.00 Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2009.01.023

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spatial resolutions and periodicities. Several factors need to be quantiﬁed in order to retrieve LST from satellite TIR data, such as, sensor radiometric calibrations (Wukelic et al., 1989), atmospheric correction (Cooper and Asrar, 1989), surface emissivity correction (Norman et al., 1990), characterization of spatial variability over land cover, and the combined eﬀects of viewing geometry, background, and fractional vegetative cover. In estimation of LST from TIR data, the digital number of the image pixel needs to be converted into spectral radiance using the sensor calibration data (Markham and Barker, 1986) and emissivity correction. However, radiance converted from digital number does not represent true surface temperature but a mixed signal or a sum of diﬀerent fractions of energy. These fractions include the energy emitted from the ground, upwelling radiance from the atmosphere, as well as the downwelling radiance from the sky, integrated over the hemisphere above the surface and type of land use/land cover surface present. These factors are dependent on atmospheric conditions and emissivity of the land surface. Therefore, the eﬀects of both atmosphere and land surface emissivity must be corrected for an accurate measurement of LST. Sobrino et al. (2001) have carried out comparative study of land surface emissivity retrieval from NOAA data and concluded that the NDVI method for land surface emissivity retrieval can be applied to obtain land surface temperature and emissivity without loosing accuracy. Atmospheric corrections for VNIR data were carried out using ‘Dark object subtraction’ algorithm. TIR data was corrected for atmospheric eﬀects using online radiative transfer codes (http://www.atmcorr.gsfc.nasa.gov/), which use MODTRAN software and other algorithms, for compensation of atmospheric transmittance, downwelling sky radiance and upwelling atmospheric radiance. The atmospheric correction parameter calculator uses the national centers for environmental prediction (NCEP) modeled atmospheric global proﬁles for a particular date, time and location as input. Several approaches, such as, single-channel algorithm, split-window algorithm, single-channel and multi-angle algorithm, and multi-channel and multi-angle algorithm have been developed in the recent past (Dash et al., 2002) for LST retrieval from TIR data. Split-window algorithms are commonly used for multi-channel TIR data (Becker and Li, 1990; Vidal, 1991; Kealy and Hook, 1993; Majumdar and Mohanty, 1998; Coll et al., 2005). Landsat ETM+ has a single TIR band (band 6), which makes the use of general split-window algorithm impossible, but the high spatial resolution of TIR band 6 in Landsat TM/ETM+ makes it suitable for local and regional TIR study. The calibration of ETM+ thermal data has been monitored over its history. A recent eﬀort has shown that the entire archive of thermal data can be calibrated to ±1 K at 300 K by incorporating a 0.7 K oﬀset error (Barsi et al., 2003). Qin et al. (2001) have used single channel algorithms to study the change in land surface temperature across the Israel–Egypt border. A single channel algorithm was suc-

cessfully used by researchers in LST retrieval (Schott and Volchok, 1985; Wukelic et al., 1989; Goetz et al., 1995; Sospedra et al., 1998; Schott et al., 2001; Weng et al., 2004) from Landsat TM/ETM+ TIR data. Unlike multithermal band systems, such as AVHRR, MODIS and ASTER, the Landsat with single thermal band does not provide any opportunity to inherently correct for atmospheric eﬀects. Ancillary atmospheric data are required to make the correction from top-of-atmosphere (TOA) radiance or temperature to surface-leaving radiance or temperature. However, with the long history of calibrated data, there is strong motivation to use these unique data for absolute temperature studies. In the present study, an attempt has been made to retrieve LST from Landsat-7 ETM+ TIR data and explore the possibility of improving LST algorithm. ‘Dark object subtraction’ algorithms considering Rayleigh scattering and atmospheric transmittance known as DOS3 (Song et al., 2001) were used for atmospheric correction on bands 3 and 4 of Landsat ETM+. These atmospheric corrected bands were used in estimating NDVI which were subsequently used in emissivity measurements. Atmospheric eﬀect on the TIR band was calculated using the available online Atmospheric Correction Parameter Calculator (http://www.atmcorr.gsfc.nasa.gov/) developed by Barsi et al. (2003). These emissivity estimate and atmospheric parameter were used in a single channel algorithm to generate surface radiance. This radiance image can then be converted to LST using a Landsat speciﬁc estimate of the Planck curve. In another approach for estimation of LST, spectral radiance was converted to brightness temperature; and emissivity correction (Artis and Carnahan, 1982) was applied later to retrieve LST. All the algorithms on LST retrieval were developed in Spatial Modeler (ERDAS, 2002). The study area, Singhbhum Shear Zone, was taken up for the present investigation due to its suitability for the technique applied and availability of ground measurement. The study area is a part of Jharkhand State of India, which lies between longitudes of 85°300 –86°300 E and latitudes 22°–23°N. It is economically rich in copper deposits. It has several land cover types, viz., dense forest, degraded forest, crop, mixed land (crop, shrub, and grass), bare soil, urban area, rock body and water body (Fig. 1). Landsat ETM+ data (path/row: 140/44) of October 26, 2001 and November 2, 2001 (overpass time: 04:34 GMT) have been acquired for the present study (Table 1). The study area falls under tropical atmosphere condition with continental aerosol type. Single channel algorithms were tested in this study using a variety of atmospheric and emissivity correction approaches for LST retrieval. LST results obtained from Landsat TIR data were validated with ground truth data and compared with ASTER and MODIS LST products on diﬀerent spatial scales. MODIS LST product (MOD11_L2) of 1 km spatial resolution which uses splitwindow algorithm (Wan and Dozier, 1996) on bands 31 and 32 and ASTER LST product (AST08) of 90 meter spa-

P.K. Srivastava et al. / Advances in Space Research 43 (2009) 1563–1574

Fig. 1. False color composite (FCC) of the study area (1: dense forest, 2: Subarnarekha river, 3: cropland, 4: mixed land, 5: rock body, 6: water body).

tial resolution which uses Temperature Emissivity Separation (Gillespie, 1998) algorithm on bands 10-14 of October 26, 2001 were obtained from Earth Observing System Data Gateway (http://www.elpdl03.cr.usgs.gov/pub/imswelcome/).

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(2) upwelling radiance from the atmosphere, and (3) downwelling radiance from the sky. The diﬀerence between the TOA brightness temperature and LST ranges generally from 1 to 5 K in the 10–12 lm spectral region, subject to the inﬂuence of the atmospheric conditions (Prata et al., 1995) and emissivity eﬀect (Van de Griend and Owe, 1993; Valor and Caselles, 1996). Therefore, atmospheric eﬀects, such as, absorption, upward emission, and downward irradiance reﬂected from the surface (Franca and Cracknell, 1994), need to be corrected. These TOA brightness temperatures should also be corrected for spectral emissivity values prior to the computation of LST to account for the roughness properties of the land surface, the amount and nature of vegetation cover, and the thermal properties and moisture content of the soil (Friedl, 2002). Two approaches have been developed to recover LST from TIR imagery (Schmugge et al., 1998). In the ﬁrst approach, a number of radiative transfer codes (RTCs), such as, LOWTRAN and MODTRAN, based on radiative transfer theory have been developed to correct for atmospheric eﬀects in satellite images (Kneizys et al., 1988; Haan et al., 1991; Vermote et al., 1997). Studies have shown that these radiative transfer codes can accurately convert the satellite measurements to surface radiance (Holm et al., 1989; Moran et al., 1992), which can be converted to LST using suitable emissivity correction algorithm. However, these corrections require accurate measurements of atmospheric properties at the time of image acquisition, which for this study were unavailable or of questionable quality, which makes routine atmospheric correction of images diﬃcult with RTCs. However, in a second approach, atmospheric and surface parameters are derived from the image itself to correct for atmospheric eﬀects, and estimate emissivity values to derive accurate LST.

2. Background 2.2. Surface reﬂectance

2.1. Retrieval of LST Satellite TIR sensors measure top of the atmospheric (TOA) radiances, which can then be converted to TOA brightness temperatures using Planck’s equation. Brightness temperature can be corrected to LST if accurate information on atmospheric condition and land surface emissivities is available. TOA radiances obtained from satellite TIR data are the result of mixing three fractions of energy: (1) emitted radiance from the Earth’s surface, Table 1 Dates and times of ground measurements and satellite data acquisitions. Date (day/ month/year)

Ground truth time (IST)

Landsat-7 time (IST)

ASTER time (IST)

MODIS time (IST)

26/10/2001 01/11/2001 02/11/2001 03/11/2001 04/11/2001

– 10:00–11:00 10:00–11:00 10:00–11:00 10:00–11:00

10:04 – 10:04 – –

10:30 – – – –

10:30 – 10:30 – –

In the present investigation, atmospheric corrections were applied on Landsat ETM+ bands 3 and 4 using dark object subtraction (DOS3) approach for estimation of surface reﬂectance (Song et al., 2001). The basic equation to convert radiance to reﬂectance, considering atmospheric eﬀects, is: qk ¼ ðp d 2 ðLsat;k Lhaze;k Þ=½svk ðESUNk cosðhz Þ szk þ Ekdown Þ

ð1Þ

where q is reﬂectance, d is Earth–Sun distance in astronomical units (AU), Lsat is at-satellite radiance, ESUN is exoatmospheric solar irradiance, hz is solar zenith angle (90 – solar elevation angle), k is subscript indicating that these values are spectral band-speciﬁc, Lhaze is path radiance = upwelling spectral irradiance due to atmospheric scattering, sv is atmospheric transmittance along the path from surface to sensor = exp(d sec hv), where d is atmospheric optical thickness, hv = 0 (viewing angle for

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nadir-looking systems), sz is atmospheric transmittance along the path from the Sun to the surface = exp(d sec hz) and Edown is downwelling spectral irradiance due to atmospheric scattering. For the study area, above parameters were calculated as d = 0.9925, hz = 43.68, sv = 0.9193, sz = 0.8902, Edown = 8.64 W/m2/lm Lhaze correction is computed using the following equation: Lk;haze ¼ Lk;min Lk;1%

ð2Þ

where Lk,min is at-satellite minimum spectral radiance, Lk,1% is radiance of black body (assuming 1% reﬂectance) Lk,min is computed from the equation: Lk;min ¼ LMINk þ QCAL ðLMAXk LMINk Þ=QCALMAX

ð3Þ

where QCAL is minimum DN, QCALMAX = 255, and constants LMINk, LMAXk are given in Landsat-7 Science Data Users Handbook (2006). Due to atmospheric scattering eﬀect, the dark object is not absolutely dark. Hence, assuming 1% reﬂectance (q = 0.01) of black body (Moran et al, 1992; Chavez, 1996), the computed radiance of a dark object is obtained using the formula, Lk;1% ¼ 0:01 ½svk ðESUNk cosðhz Þ szk þ Ekdown Þ=ðp d 2 Þ

ð4Þ

DOS3 computes sv as ed= cosðhv Þ and sz as ed= cosðhz Þ assuming Rayleigh scattering only. Edown parameter is calculated using 6S radiative transfer codes for standard atmosphere (Vermote et al., 1997) for a Rayleigh atmosphere; that is, zero aerosol at optical depth 550 nm. hv is assigned to be 0 for nadir looking Landsat ETM+ satellite. The optical thickness for Rayleigh scattering (d) is estimated following Kaufman (1989) as, d ¼ 0:008569k4 ð1 þ 0:0113k2 þ 0:00013k4 Þ

ð5Þ

where k is wavelength in lm. QCALMAX, LMINk, LMAXk, hz and date of image acquisition are obtained from image header record; d (Earth–Sun distance in astronomical unit) is generated online at HORIZONS Web Interface (2007), while ESUNk is obtained from Landsat7 Science Data Users Handbook (2006). 2.3. Emissivity measurement The knowledge of surface emissivity is important in reducing the error in estimation of LST from satellite data. Lack of knowledge of emissivity can introduce an error ranging from 0.2 to 1.4 K for an emissivity of 0.98 and at the ground height of 0 km, when a single channel method of LST estimation is used (Dash et al., 2002). Surface emissivity can be measured directly through instrument or can be estimated through various available algorithms. In the present study, emissivity measurement is obtained from NDVI based method. The eﬀect of land surface emissivity on satellite measurements can be generalized into three cat-

egories: (i) reduction of surface-emitted radiance; (ii) nonblack surfaces reﬂect radiance; and (iii) the anisotropy of reﬂectivity, which may reduce or increase the total radiance from the surface. There are a number of emissivity measurement techniques for multi-band thermal data, such as, normalized emissivity method (Gillespie, 1985), thermal spectral indices (Becker and Li, 1990), spectral ratio method (Watson, 1992), alpha residual method (Kealy and Gabell, 1990; Hook et al., 1992), classiﬁcation-based estimation (Snyder et al., 1998), and the temperature emissivity separation method (Gillespie et al., 1998). These approaches are successfully applied in multi-TIR bands but not feasible in the present study due to single TIR band in Landsat ETM+. It is important to carry out atmospheric correction before deriving NDVI for emissivity estimation because atmospheric eﬀect contaminates NDVI signal (Song et al., 2001) and the modiﬁcation is nonlinear. It is clear from the comparison of diﬀerent corrected and uncorrected land cover categories that NDVI signal is nonlinear. NDVI value is underestimated in uncorrected dense forest, forest and mixed land cover, while they are similar in bare soil, crop and rock body. On the other hand, NDVI is overestimated in uncorrected water bodies. DOS3 algorithms (Song et al., 2001) were used to obtain atmospherically corrected reﬂectance image. In the present study, NDVI based emissivity estimation method (Valor and Caselles, 1996) has been used, because of its simpliﬁed approach and better accuracy. The study area has mixed land cover type of soil and vegetation cover, and nearly isothermal media during the time of image acquisition on November 2, 2001, which satisfy the conditions of Valor’s emissivity estimation approach. The proportion of vegetation cover, Pv, for each pixel through satellite data is calculated using the following relationship (Valor and Caselles, 1996): Pv ¼

ð1 i=is Þ ð1 i=is Þ kð1 i=iv Þ

ð6Þ

2v q1v Þ where k ¼ ðq ðq2s q1s Þ where i, is and iv are NDVIs of mixed pixel, pure soil pixel and pure vegetation pixel respectively; q2v and q1v are reﬂectances in the NIR and red regions for pure vegetation pixel; q2s and q1s are the reﬂectances in the NIR and red regions for pure soil pixel, and q2 and q1 are reﬂectances measured in NIR and red bands for mixed pixels, respectively. According to Rouse et al., (1974), NDVI is deﬁned as,

i¼

q2 q1 q2 þ q1

ð7Þ

NDVI value of mixed pixel is obtained through Eq. (7) and substituted in Eq. (6) to get proportion of vegetation (Pv). NDVI of a mixed pixel (i) can also be calculated using the equation suggested by Valor and Caselles (1996) as, i ¼ iv P v þ is ð1 P v Þ þ di

ð8Þ

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where di is the correcting factor obtained (Valor and Caselles, 1996) as: di ¼

P v ðq2v q1v Þ þ ð1 P v Þðq2s q1s Þ iv P v þ is ð1 P v ðq2v þ q1v Þ þ ð1 P v Þðq2s þ q1s Þ P vÞ

ð9Þ

Emissivity (e) is a function of bare soil and vegetation, related as follows: e¼

ev es es ðiv þ diÞ ev ðis þ diÞ iþ þ de iv is iv is

ð10Þ

where de ¼ 4ðDeÞP v ð1 P v Þ

ð11Þ

In the above Eqs. (10) and (11), es and ev are the respective emissivities of bare soil and pure vegetation, de is internal reﬂection emissivity due to cavity eﬀect, and De is the weighted mean of de. The area is covered by about 15% forest, 24% vegetation land, 57% mixed land, 2% bare soil and 2% water body. Based on land cover type, weighted mean emissivity (De) is approximated to be 0.006 (Valor and Caselles 1996; Rubio et al., 1997). Emissivity (e) of the area is obtained from Eq. (10) after solving Eq. (7) for NDVI, which is substituted in Eq. (6) to get Pv. Pv is used in Eq. (9) to solve di, and in Eq. (11), for de. Other mentioned variables, viz., es and ev, and is and iv are obtained from literature (Valor and Caselles, 1996; Rubio et al., 1997) and image histogram. There were no ﬁeld measurements available for bare soil and pure vegetation emissivity values. Mean emissivity values of bare soil as 0.960 and vegetation as 0.985 (Nerry et al., 1990; Schmugge, 1990; Valor and Caselles, 1996; Rubio et al., 1997) were considered for the study area, in the 10.5 to 12.5 lm spectral range. Emissivity value of 0.989 (Snyder et al., 1998) was taken for water body. These values were used in Eq. (10) to estimate emissivity for the whole study area. Land surface temperatures evaluated from emissivity measurements obtained using Valor and Caselles (1996) formula (Eqs. (6)–(11)) were compared subsequently with that obtained using Van de Griend and Owe (1993) formula (Eq. (12), given below) developed for 10 diﬀerent surfaces. However, the formula proposed by the latter is applicable for the NDVI values ranging from 0.175 to 0.727 with an estimated error of 0.7% in the emissivity. Since about 98% of study area falls within the presumed Van de Griend and Owe (1993) NDVI ranges, the comparison of the results was done over that 98% area. The relationship between NDVI and emissivity, as proposed by Van de Griend and Owe (1993) is given as: e ¼ 1:0094 þ 0:047 lnðNDVIÞ

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range from 0.985 to 0.989, while those using the Van de Griend and Owe (1993) method range from 0.99 to 0.994. Higher values of emissivity in Van de Griend and Owe (1993) method are above the maximum value of emissivity of vegetation, which is 0.99; so this method overestimates emissivity values for fully vegetated pixels. On the other hand, in the case of bare soil/mixed land, it is observed that the emissivity values obtained by Van de Griend and Owe (1993) method is lower (<0.96) than that obtained by Valor and Caselles (1996) (0.96–0.975) method; the latter value is close to the emissivity values for bare soil/mixed land. Therefore, it indicates that Van method underestimates bare soil/mixed land emissivity. These observations were checked with ground data to support the interpretation (Fig. 2). So we have used emissivity results obtained from Valor and Caselles (1996) for the rest of the analysis. 2.4. Single channel algorithm for LST retrieval To evaluate atmospheric eﬀect and emissivity correction on the Landsat ETM+ TIR band, a single channel algorithm was adopted. Lk without considering atmospheric eﬀect is estimated from Eq. (13). Lk, the thermal infrared radiance received by satellite sensors, is mainly composed of three parts; namely, radiance by surface, reﬂected downward radiance from atmosphere and upward radiance from atmosphere (Barsi et al., 2003); as given by Eq. (14): ðLMAXk LMINk Þ ðQCALMAX QCALMINÞ ðQCAL QCALMINÞ þ LMINk

ð13Þ

Lk ¼ ½ek Lk ðT s Þ þ ð1 ek ÞLkdown s þ Lkup

ð14Þ

Lk ¼

ð12Þ

Emissivity values obtained in the above two discussed methods were correlated with NDVI value of the study area derived earlier. Fully vegetated areas are having NDVI values more than 0.6. The area with higher values of NDVI, which corresponds to vegetation, shows that Valor and Caselles (1996) method derived emissivity values

Fig. 2. Plot of Valor emissivity versus Van emissivity in mixed lands and pixels covered by vegetation.

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where s is the atmospheric transmission; e is the emissivity of the surface; Lk(Ts) is the radiance of a blackbody target of kinetic temperature Ts; Lkup is the upwelling or atmospheric path radiance; Lkdown is the downwelling or sky radiance; and Lk is the space-reaching radiance measured by the sensor. As there were no meteorological data at the time of satellite overpass, the atmospheric parameters required for Eq. (13) were obtained from online Atmospheric Correction Parameter Calculator (http://atmcorr.gsfc.nasa.gov/) developed by NASA for Landsat TM/ETM+ TIR data. Calculations of required parameters have been done for the overpass date and time of the satellite data acquired at several places in the study area (October 26, 2001 and November 2, 2001; overpass: 4:34 GMT). The averaged values of these calculated parameters are: is 3.58 Wm2 lm1 Sr1, Lkup is 2.48 Lkdown 2 1 Wm lm Sr1 and s is 0.67 for October 26, 2001 and is 4.74 Wm2 lm1 Sr1, Lkup is 3.1 Lkdown 2 1 1 Wm lm Sr and s is 0.62 for the November 2, 2001. Lk was then calculated from Eq. (13), and the land surface radiance, Lk (Ts) of kinetic temperature Ts can be calculated as follows: Lk Lkup 1 ek Lkdown sek ek

Lk ðT s Þ ¼

ð15Þ

Land surface radiances derived from Landsat ETM+ TIR data were subsequently converted into LST using the following relationship that is similar to the Planck equation with two pre-launch calibration constants (Schott and Volchok, 1985; Wukelic et al., 1989): Ts ¼ ln

K2 K1

LkðT s

þ1 Þ

ð16Þ

where Ts is the LST in Kelvin which is changed to degree Celsius (°C) by subtracting 273.13; K1 and K2 are ETM+ thermal constants, whose values are 666.09 Wm2 lm1 Sr1 and 1282.71 K (LANDSAT-7 Science Data Users Handbook, 2006), respectively.The emissivity corrected LST were also computed using Artis and Carnahan (1982) formula. The Lk value was obtained from Eq. (13), and substituted in Eq. (16) to derive eﬀective at-satellite temperature (TB) in Kelvin. The temperature obtained from Eq. (16), is the at-satellite temperature (TB) unlike previous LST (Ts) because the at-satellite radiance value is used instead of the emissivity and atmospheric corrected radiance value. LST (Ts) was then computed following (Artis and Carnahan, 1982) equation: Ts ¼

TB 1 þ ðk T B =qÞ ln e

ð17Þ

where k is the wavelength of emitted radiance. k = 11.5 lm (Markham and Barker, 1986) and q = h c/r = 14380 m K. Here, r is Boltzmann constant (1.38 * 1023 J/K), h is Planck’s constant (6.26 * 1034 Js) and c is velocity of light (2.998 * 108 m/s).

The two NDVI based emissivity measurement results were used in LST retrieval (Artis and Carnahan, 1982; Barsi et al., 2003) approaches on low gain and high gain modes of Landsat ETM+ band 6. Spectral radiance values were derived from low gain and high gain modes of Landsat ETM+ TIR band and compared. The diﬀerence in low and high gain modes are found to be within ±0.14, mean is 0.004, standard deviation is 0.038 and median is 0.00056. It shows that both the bands are identical, and hence, taking either band for study of LST, will not aﬀect the results. In the present investigation, low gain mode TIR band 6 of Landsat ETM+ was taken for LST retrieval. 3. LST validation One of the major problems in the validation of remote sensing data with ground truth observation is the dissimilarity between the spatial scales of ﬁeld thermometers (<1 m2) and that of satellite sensors (60 60 m2 for Landsat-7 ETM+ TIR). The comparison of ground (point) measurements with that of satellite (area averaged) data is meaningful only when the test site is homogeneous in both temperature and emissivity at various spatial scales involved. Accuracy of ground measurements must be assessed, including the natural variability of the surface. Ideal validation of a test site is very diﬃcult to achieve; however, dense vegetation, water body and bare soil viz., dry lakes or playas, have been used as most suitable sites (Coll et al., 2005) for validation. Thermal homogeneity test of the various land use classes present in the study area, namely, dense forest, degraded forest, crop, mixed land, bare soil, rock body and water body were carried out. It was observed that dense forest, mixed land and water body have uninterrupted large areal extension (>1 km2) hence are thermally homogeneous, while other classes have intermixing and only small uninterrupted extensions (<0.1 km2) with little thermal homogeneity. 3.1. Satellite data Low gain band 6 of Landsat-7 of November 2, 2001 was selected for validation of LST results (Suga et al., 2003; Weng et al., 2004) obtained by using equations 16 and 17 after considering surface emissivity values (Valor and Caselles, 1996). Thermal homogeneities of the land use classes at measurement sites were assessed at the Landsat-7 spatial scales after evaluating atmospheric corrected LST (Barsi et al., 2003). For each land use class, the LSTs for the pixels closest to each measurement site were extracted, by selecting arrays of 55, 99 and 1717 pixels centered around the target pixel, for calculating the average temperature (Tav), standard deviation (r), minimum (Tmin) and maximum (Tmax) temperatures for diﬀerent classes. Low r is indicative of least spatial variation and hence thermally homogeneous class, while high r is for signiﬁcant change of temperature and thus low homogeneity. The spatial scale of 17 17 pixels (equivalently, 1 1 km2) was taken

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to compare the result with MODIS LST (MOD11_L2) of 1 km spatial resolution obtained from EOS data gateway. In the present study, thermal homogeneity was observed only over dense forest, water body and to some extent over mixed land at target sites. However, due to smaller size (<0.10 km2) of bare soil at target site, only 5 5 pixel sizes could be tested for homogeneities. Thermal homogeneity of rock body and crop could not be established at target sites due to their small areal coverages. Results of thermal homogeneity study show (Table 2) that while maximum diﬀerences of Tav between diﬀerent arrays are 0.32 °C and 0.40 °C respectively for dense forest and water body, the ranges of standard deviations (r) for these two classes are from 0.32 °C to 0.34 °C and 0.37 °C, respectively. Low values of diﬀerences of Tav between diﬀerent arrays and r are indicative of thermal homogeneity over water body and dense forest. However, for mixed land, the maximum differences of Tav in arrays and that of r ranges were 0.41 °C to 0.63 °C, respectively. Mixed land may also be taken for validation due to low value of maximum diﬀerence between Tav in arrays. In case of the bare soil which is of size <0.10 km2, r is 1.62 °C and diﬀerence of maximum and minimum temperature is 6.3 °C in 5 5 arrays of pixels. High spatial variation of temperature in bare soil at target site makes it unsuitable for comparison with ground measurements, because no thermal homogeneity is maintained; while thermal homogeneities of crop and rock body could not be tested due to their smaller sizes (<0.01 km2). On the other hand, thermal homogeneities observed over dense forest, water body and mixed land based on Landsat-7 data of November 2, 2001, were taken for October 26, 2001 also, because there is no change in land use classes over temporal scale of 7 days. Based on thermal homogeneities, ground temperature data obtained over dense forest, water body and mixed land were compared with satellite data, while other classes were left out

Table 2 LST of Landsat-7 for arrays of 1 1, 5 5, 9 9 and 17 17 for diﬀerent classes of November 2, 2001. Land use class

Arrays

Tav (°C)

r (°C)

Tmin (°C)

TMax (°C)

Dense forest

11 55 99 17 17

27.13 26.87 26.86 26.81

– 0.34 0.33 0.32

– 26.55 26.52 26.52

– 27.33 27.33 27.33

Water body

11 55 99 17 17

28.41 28.84 28.83 28.81

– 0.37 0.37 0.37

– 28.39 28.39 28.39

– 29.20 29.20 29.21

Mixed land

11 55 99 17 17

32.49 32.10 32.08 32.12

– 0.41 0.59 0.63

– 31.58 30.99 30.99

– 32.89 33.29 33.56

Bare soil

11 55 99 17 17

37.25 37.31 – –

– 1.62 – –

– 33.97 – –

– 40.27 – –

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due to lack of their thermal homogeneities. Landsat-7 LST results were also compared with LST data products of ASTER (AST08) and MODIS (MOD11_L2) of the same dates. 3.2. Ground measurements Surface temperature measurements were carried out over dense forest, crop, mixed land, bare soil, rock body and water body during daytime, cloud-free concurrently to the overpasses times of Landsat and MODIS sensors, between 10:00–11:00 IST (GMT+5:30). Contact thermometer and AG-42D Model Infrared Thermometer were used for collecting ground surface temperatures. AG-42D Model of 0.1 °C resolution measures ground temperature in single channel (8–14 lm) with an accuracy of ±0.5 °C. It has an emissivity control panel to calibrate the radiation characteristic of the surface body, being measured in the emissivity range 0.01–0.99 with resolution of 0.01. Surface temperatures measured using AG-42D Thermometer were corrected for emissivity eﬀects by adjusting the emissivity control panel to the emissivity values of ground surface (Salisbury and D’Aria, 1992). In order to capture the spatial variability of the surface temperature within the land use class at measured site, several readings were recorded at intervals of about 100 m around the site. Average ground temperatures over the land use classes at various measured sites were estimated with maximum standard deviation of <0.5 °C for all locations. Contact thermometer was used simultaneously to measure the ambient temperatures of all the land use classes, which were compared with ambient temperatures measured by Model to check its accuracy. The diﬀerences in the measured ambient temperatures by these two measurements were found to be less than 0.30 °C. Contact thermometer was not able to measure correct surface temperatures of land use classes except for water body because of lack of proper contact with surface. Surface temperatures of water body measured during ﬁeld work using contact thermometer and AG-42D Model were compared and found to be within ±0.2 °C. Temporal variations of temperatures over water body and mixed land during the ﬁeld observations were considered and were found to be less than 1 °C. Ground measurements recorded over thermally homogeneous classes of dense forest, water body and mixed land were compared with LST results of Landsat-7 TIR data of November 2, 2001, while that of crop, bare soil and rock body were not compared with Landsat-7 LST because of lack in their thermal homogeneities. LST results derived from Landsat-7 TIR data of November 2, 2001, considering atmospheric eﬀects, were compared with emissivity corrected ground temperatures measurements over-dense forest, water body and mixed land (Table 3 and Fig. 3). LST results over dense forest and water body derived using Eqs. (16) and (17), with as well as without atmospheric correction parameters of Landsat-7 TIR data, were found to be within ±2.0 °C of the ground measurement results. LST results derived over

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Table 3 Comparison of ground measured temperature with Landsat-7 ETM + LST. Land use class

LST ± r (°C) (Eq. (16)) with atmospheric correction

LST ± r (°C) (Eq. (16)) without atmosphere correction

LST ± r (°C) (Eq. (17)) with atmospheric correction

LST ± r (°C) (Eq. (17)) without atmospheric correction

Ground LST±r (°C) (AG-42D)

Dense forest Water body Mixed land

27.80 ± 0.33

24.56 ± 0.28

27.84 ± 0.33

24.60 ± 0.28

26.30 ± 0.21

28.26 ± 0.37

25.32 ± 0.30

28.29 ± 0.37

25.35 ± 0.30

26.70 ± 0.16

32.34 ± 0.63

28.13 ± 0.46

32.40 ± 0.63

28.19 ± 0.46

33.10 ± 0.29

3.3. Data comparison

Fig. 3. Comparison of estimated land surface temperature (LST) with or without atmospheric (ATM) parameters over diﬀerent land use classes with ground LST; error bars indicate uncertainty (r) in LST measurements.

dense forest and water body, using Eq. (17) (Artis and Carnahan, 1982) without atmospheric correction, diﬀer by 1.50 °C from ground measurement results. However, considering the accuracy of AG-42D Model Thermometer (±0.50 °C) and uncertainty in LST measurements (LST ±r), we can not be sure about the better performance of LST estimation (Eq. (17)) without atmospheric parameters. LST result obtained over mixed land shows much diﬀerence with observed temperature when we do not consider atmospheric parameters. Here diﬀerence in LST between observed and derived temperature with atmospheric parameters are within the range of 1 °C, and in the range of 5.0 °C without atmospheric parameters. Atmospheric parameters are must for retrieving accurate LST over different land use classes.

Reliability of LST results derived from TIR sensor may not always be possible to validate with ground surface temperature due to non-availability of ﬁeld data concurrently with satellite overpass. LST of the study area can be retrieved from other TIR sensors and compared on spatial scales with Landsat-7 LST if their overpass time is close to Landsat-7 on a given day. Diﬀerent algorithms and atmospheric parameters had been used to derive LST from various TIR sensors. A multi-sensor approach will provide us more frequent and accurate monitoring of study area, if we get same LST retrieved from various sensors on spatial scales over thermally homogeneous sites and validate with ground measurements. Reliability of such LST derived for other date where ground measurements are not available, will be more if we get same LST through diﬀerent TIR sensors. In this study, ASTER and MODIS TIR sensor data were also used for LST comparison and validation with ground measurements. LST data products of MODIS (MOD11_L2) of 1 km spatial resolution which uses splitwindow algorithm (Wan and Dozier, 1996) on bands 31 and 32, and ASTER (AST08) of 90 meter spatial resolution which uses Temperature Emissivity Separation (TES) algorithm on bands 10–14 were obtained from EOS data gateway (http://www.elpdl03.cr.usgs.gov/pub/imswelcome/) for comparison (Table 1). MODIS LST (Wan and Dozier, 1996) and ASTER LST (Gillespie et al., 1998) data products use equation 16 along with atmospheric parameters in their algorithms for LST retrieval; these LST results were compared with the LST obtained from Landsat-7 data using equation 16 along with atmospheric parameters (Barsi et al., 2003). LST results of the thermally homoge-

Table 4 Atmospheric corrected LST comparison of TIR sensors with ground truth results. Date (day/month/year)

Land use class

Landsat-7 LST ± r (°C) (Eq. (16))

MODIS LST ± r (°C)

ASTER LST ± r (°C)

Ground LST ± r (°C)

26/10/2001

Dense forest Water body Mixed land

29.10 ± 0.43 28.71±0.23 31.70±0.73

28.42±0.31 28.22±0.11 32.39±0.45

28.24±0.33 28.68±0.08 31.30±0.82

– – –

02/11/2001

Dense forest Water body Mixed land

27.80±0.33 28.26±0.37 32.34±0.63

27.84 ±0.23 28.41±0.18 32.46±0.35

– – –

26.30±0.21 26.70±0.16 33.10±0.29

P.K. Srivastava et al. / Advances in Space Research 43 (2009) 1563–1574

neous classes of dense forest, water body and mixed land obtained from diﬀerent sensors along with ground truth is presented in Table 4. LST results obtained from Landsat-7 and ASTER were considered on 1 km spatial scale, while 3 3 arrays were taken for LST results obtained with MODIS over thermally homogeneous classes. Comparative analyses of all these LST values obtained using multi-sensor data on the dates considered in this study show that all the LST values are within the range of 2 °C vis-a`-vis ground measurements on November 2, 2001. This diﬀerence may be due to inaccurate assessment of atmospheric parameters and instrumental errors. Higher variability (r) in Landsat-7 derived LST, in comparison with that from MODIS and ASTER data, for each land use class is more likely due to its smaller ground sampled distance (GSD) allowing it to study the real variation in the surface temperature. 4. Results and discussion Various algorithms were used on cloud-free Landsat-7 ETM+ image of Singhbhum Shear Zone region, Jharkhand, India for LST retrieval. Atmospheric eﬀects on satellite data channels 3 and 4 of ETM+ were removed using dark object subtraction (DOS3) approach, accounting for atmospheric transmittance and downwelling diﬀuse irradiance due to Rayleigh scattering eﬀect. In this study, NDVI is obtained from atmospheric corrected reﬂectance of bands 3 (0.66 lm) and 4 (0.84 lm). Emissivity measure-

Fig. 4. Emissivity measurements using Valor and Caselles (1996) algorithm.

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ments using Valor’s (Eq. (9)) and Van’s (Eq. (12)) formulae were used in this study. Emissivity values vary from 0.95 to 0.99 (Fig. 4). High emissivity values are corresponding to dense forest, vegetation and water body, while low values correspond to residential complex, rock body and bare soil (Figs. 1 and 4). A close study of Fig. 2 shows that Van’s approach has overestimated emissivity values at higher ranges, while underestimating the same at lower ranges when compared with the results obtained with Valor’s approach. Valor’s emissivity had been used for rest of the study in LST estimation due to its better accuracy as discussed in Section 2.3. A direct relationship was observed in the emissivity range between 0.97–0.985 and surface reﬂectance values in band 3 (Fig. 5) from 32 pixel samples spread over the area. These emissivity values correspond to mixed land and pixels covered by vegetation. Thermal homogeneity over land use classes of dense forest, water body and mixed land were observed due to their un-interrupted large areal extension (>1 km2) at measurements sites. However, thermal homogeneity over crop, bare soil and rock body could not be observed due to their small areal coverages (<0.10 km2) and intermixing of classes on spatial scales (Table 2). Thermally homogeneous classes were taken for LST comparison with ground temperature measurements, while non-homogeneous classes were considered unsuitable for LST comparison (Table 3). Four approaches, namely: (i) Artis and Carnahan (1982) equation with atmospheric correction; (ii) Artis and Carnahan (1982) equation without atmospheric correction; (iii) Barsi et al. (2003) approach with atmospheric correction and (iv) Barsi et al. (2003) approach without atmospheric correction were used to derive true land surface temperature (Fig. 3) of November 2, 2001 Landsat-7 data, and compared with ground truth measurements over thermally homogeneous classes. It was observed that Artis and Carnahan (1982) equation (Eq. (17)) without atmospheric correction is close to ground measurements in dense forest and water body but in mixed land there is discrepancy of temperature about 5 °C (Table 2). LST results obtained using Eqs. (16) and (17) with atmospheric correction gives better approximation in all the homogeneous classes. Large diﬀerence of temperature between LST results retrieved

Fig. 5. Plot of emissivity versus surface reﬂectance for Landsat ETM+ band 3.

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without atmospheric corrections and ground temperature measurements in mixed land shows that atmospheric correction is a must. High uncertainty (r) in atmospheric corrected LST with respect to non-atmospheric corrected LST is probably due to cross variability within atmospheric transmission (which is less than 1) reducing the range in the radiance at the sensor. LST data products of ASTER (AST08) and MODIS (MOD11_L2) obtained from EOS data gateway (Table 4) for same dates of Landsat-7 LST results were compared with ground measurements. ASTER and MODIS data products use TES algorithm and splitwindow algorithm respectively considering atmospheric parameters for each pixel. LST results (Barsi approach) derived from Landsat-7 data of October 26, 2001 and November 2, 2001 are very close to ASTER and MODIS data products over the homogeneous classes of dense forest, water body and mixed land. LST results obtained from multi-sensor data of Landsat-7 and MODIS yielded closer to ground temperature measurements over dense forest, mixed land and water body classes for November 2, 2001. LST diﬀerence over land use classes of these two sensors were within 2 °C. Further comparison of LST results of Landsat-7 and MODIS with ASTER of October 26, 2001 again yielded LST within a temperature diﬀerence of <2 °C. These results indicate that multi-sensor approach can be applied to any study area with more reliability of LST results, if they all show similar LSTs over diﬀerent classes present in the study area. Finally, land surface temperature (LST) image of the entire study area (Fig. 6) using best approximate of Valor

emissivity value in Artis and Carnahan (1982) equation along with atmospheric correction shows that LST values vary from 23 to 40 °C. Lower values correspond to dense vegetation and water bodies (Fig. 1), while higher values are observed over residential complexes, bare soils and rock bodies. 5. Conclusions This study shows a good agreement of LST derived from Landsat-7 ETM+ TIR data with ground temperatures over thermally homogeneous classes of dense forest, mixed land and water body. LST results from diﬀerent sensors of a given date vary within ±2 °C with ground temperature measurements. This study is the ﬁrst attempt of using ‘Dark object subtraction’ algorithm with Valor emissivity estimation in single channel algorithms. Thermal homogeneity of the measurement sites at diﬀerent spatial scale is must for comparison of ground LST with satellite data. Atmospheric parameters must be considered for LST retrieval otherwise LST results derived though satellite data may have error up to 5 °C with actual ground measurements. This integrated approach using emissivity values obtained through NDVI method in single channel algorithm is found to be the appropriate method for LST retrieval. The accuracy of the result was quite good in most of the area, with diﬀerence of ±2 °C with actual ground temperature measurements. LST results of multi-sensor data of diﬀerent dates over various land use classes are within the accuracy of 1 °C. A multi-sensor approach for LST retrieval is quite useful for frequent monitoring of any study area with greater reliability with selective ground temperature measurements. The accuracy of LST retrieved for satellite data can be further improved with accurate measurements of surface emissivity and estimates of atmospheric parameters at each pixel. Multi-sensor approach can be used for any other study area if thermally homogeneity is observed, as in the case of present study. However, for diﬀerent atmosphere, it might not work as well because it is in essence using global atmospheric parameters. Acknowledgements The authors are thankful to Global Land Cover Facility for providing Landsat ETM+ data. The ﬁrst author thanks ISRO for providing research grant to carry out the present study. Critical comments and suggestions made by the anonymous referees and Dr. P.A. Shea, Editor-in-Chief, ASR for improvement of the manuscript are also acknowledged. References

Fig. 6. Estimated land surface temperature over the study area.

Artis, D.A., Carnahan, W.H. Survey of emissivity variability in thermography of urban areas. Rem. Sens. Environ. 12, 313–329, 1982. Atmospheric Correction Parameter Calculator. Available from: http:// www.atmcorr.gsfc.nasa.gov/ (Last accessed on March 21, 2007).

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Available online at www.sciencedirect.com

Advances in Space Research 43 (2009) 1575–1587 www.elsevier.com/locate/asr

Low-latitude geomagnetic response to the interplanetary conditions during very intense magnetic storms R. Rawat *, S. Alex, G.S. Lakhina Indian Institute of Geomagnetism, Plot No. 5, Sector-18, New Panvel (W), Navi Mumbai 410 218, India Received 25 April 2007; received in revised form 22 January 2009; accepted 23 January 2009

Abstract The variations in the horizontal and declination components of the geomagnetic ﬁeld in response to the interplanetary shocks driven by fast halo coronal mass ejections, fast solar wind streams from the coronal hole regions and the dynamic pressure pulses associated with these events are studied. Close association between the ﬁeld-aligned current density (jk) and the ﬂuctuations in the declination component (DDABG) at Alibag is found for intense storm conditions. Increase in the dawn-dusk interplanetary electric ﬁeld (Ey) and DDABG are generally in phase. However, when the magnetospheric electric ﬁeld is directed from dusk to dawn direction, a prominent scatter occurs between the two. It is suggested that low-latitude ground magnetic data may serve as a proxy for the interplanetary conditions in the solar wind. Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Magnetic storms; Interplanetary magnetic ﬁelds; Solar wind

1. Introduction Geomagnetic storms are caused mainly by solar wind transients from the coronal mass ejections (CMEs) and solar ﬂares or by the corotating interaction regions (CIRs) formed during the interaction between the high and lowspeed streams. Occurrence frequency and intensity of transient solar emissions vary with diﬀerent phases of the solar cycle characterized by the number of sunspots on the photosphere. Solar maximum is dominated by powerful solar eruptions, like solar ﬂares and CMEs. On the other hand, the solar minimum is featured by coronal holes and fast wind streams (Tsurutani and Gonzalez, 1998). Consequently, the geomagnetic response to diﬀering solar conditions also varies to a wide extent. The dynamic interaction of solar wind with the magnetosphere has been studied in great detail for several decades (Burton et al., 1975; Baker et al., 1983; Gonzalez and

*

Corresponding author. Tel.: +91 22 2748 4060. E-mail address: [email protected] (R. Rawat).

Tsurutani, 1987; Gonzalez et al., 1994, 1999, 2001). Most dominant mechanism for transfer of solar wind energy into the magnetosphere to produce the geomagnetic storms is magnetic reconnection between southwardly oriented IMF Bz component and the antiparallel geomagnetic ﬁeld lines (Dungey, 1961; Akasofu, 1981; Gonzalez et al., 1989, 1994; Kamide et al., 1998). During the magnetic reconnection process, the magnetic energy gets transformed into plasma kinetic energy, which eventually is redistributed within the various regions of the magnetosphere. Several identiﬁed current systems in the magnetosphere which are always present in the magnetosphere include magnetopause current, tail current, ring current and ﬁeld-aligned (Birkeland) currents. During the magnetic storm conditions these currents get enhanced. Primarily, the intensity of magnetic storm is ascertained by the ring current, which is a westward toroidal current formed by drifting of ions and electrons in the 20–200 keV energy range between 2 and 7RE (Singer, 1957; Williams, 1987). Measure of the ring current is given in terms of disturbance storm time (Dst) index (Sugiura, 1964). The Dst index represents the axially symmetric disturbance magnetic ﬁeld

0273-1177/$36.00 Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2009.01.025

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at the dipole equator on the Earth’s surface. The strength of this disturbance ﬁeld is (approximately) given by the Dessler–Parker–Sckopke relationship (Dessler and Parker, 1959; Sckopke, 1966). At the time of storms, the ring current enhancement induces a magnetic ﬁeld opposite to the ambient geomagnetic ﬁeld (north to south) and thus causes decrease in geomagnetic ﬁeld (Chen et al., 1993; Daglis et al., 1999; Daglis and Kozyra, 2002). Dessler and Parker (1959) and Sckopke (1966) stated that the resulting ﬁeld decrease is directly proportional to the total kinetic energy of the ring current particles. It is known that the disturbance ﬁeld is generally not axially symmetric. The asymmetric disturbance ﬁeld has usually been attributed to a partial ring current (Akasofu and Chapman, 1964; Cahill, 1966; Frank, 1970; Fukushima and Kamide, 1973). Crooker and Siscoe (1981) suggested that the asymmetric disturbance ﬁeld may be produced by a net Birkeland current ﬂowing into the ionosphere near noon and ﬂowing out near midnight. In the asymmetric disturbance ﬁeld, the ﬁeld decrease is usually largest in the dusk sector. Several studies have been done to identify the solar and interplanetary origin of the geomagnetic storms (Tsurutani et al., 1988; Gonzalez et al., 1999, 2002; Zhang et al., 2007). The solar ejecta or the driver gas has two principal regions of intense magnetic ﬁelds which are causative of intense geomagnetic storms. The primary part of the driver gas contains magnetic cloud structure (Klein and Burlaga, 1982). The magnetic clouds are characterized by slowly varying and strong magnetic ﬁeld strengths (10–25 nT or higher), smooth rotation in the magnetic ﬁeld from large southern (northern) directions to large northern (southern) abnormally low proton temperatures and low proton beta (0.1) at 1 AU (Tsurutani and Gonzalez, 1995). Another region containing intense southward IMF ﬁelds is found in the sheath region, following the interplanetary shocks. Sheath ﬁelds are the compressed solar wind downstream of the interplanetary shocks. The fast interplanetary shocks are driven by interplanetary coronal mass ejections (ICMEs), which are coronal mass ejections coming from the Sun into the interplanetary space and contain high magnetic ﬁelds. Two of the several mechanisms leading to southward component ﬁelds in the sheath are, shock compression and draping. In the former mechanism, the shock compresses both the magnetic ﬁeld and plasma, while in the latter, draping of magnetic ﬁelds around the solar ejecta leads to a squeezing of plasma (Tsurutani et al., 1992; Gonzalez et al., 1999, 2007; Zhang et al., 2007 and Echer et al., 2008). Field-aligned currents (Birkeland, 1908; Zmuda et al., 1966) play important role in the solar wind–magnetospheric interaction by coupling the magnetospheric and ionospheric plasmas electrodynamically, and transferring the energy derived from the solar wind to the ionosphere ultimately. The magnetic signature of the ﬁeld-aligned currents recorded by ISIS-2 and TRIAD satellites (Zmuda and Armstrong, 1974) showed westward magnetic ﬁeld extending over 2–3 degrees of lat-

itude (200–300 km). These currents are always present. With increasing disturbance level, they intensify and move equatorward. The Crooker and Siscoe (1981) model and Harel et al. (1981) model explained the association of region 1 and region 2, Birkeland currents in the development of low latitude dawn-dusk asymmetry. Rich et al. (1990) studied the evolution of the high latitude precipitation and ﬁeldaligned currents in response to changes in the Bz component of the interplanetary magnetic ﬁeld. Chun and Russell (1997) suggested a strong association of inner magnetospheric ﬁeld-aligned currents (FACs) with low latitude ionospheric currents for varied levels of geomagnetic activity. The present work focuses on the variation of horizontal (H) and azimuthal (declination, D) components of the geomagnetic ﬁeld in response to the diﬀering solar wind and interplanetary conditions during four magnetic storm events of solar cycle-23. We attempt to investigate the correlation between the declination (D) component at low latitude observatory, Alibag with the interplanetary electric ﬁeld, Ey and variations in the ﬁeld-aligned currents as evidenced from the changes in the azimuthal component of the Earth’s magnetic ﬁeld over the low latitude location. 2. Data set and computation In order to examine the diﬀering characteristics of geomagnetic ﬁeld variations evidenced in response to changing interplanetary and solar wind conditions, during diﬀerent phases of solar cycle, four intense storm events are considered for the present analysis with Dstmin 6 200 nT. The storm events of 6 April 2000 and 31 March 2001 occurred during high solar activity period while, 20 November 2003 and 24 August 2005 occurred during descending phase of solar cycle. Another important reason to select these events is the availability of complete data set, which assisted in a detailed analysis. Solar ﬂare and CMEs information is provided, respectively, from GOES-8 satellite and LASCO instrument onboard SOHO satellite. The interplanetary and solar wind conditions for the storm development are examined through the data set acquired from MAG and SWEPAM instruments on ACE satellite upstream at L1 point (240RE). Parameters obtained from ACE includes, By, Bz components of interplanetary magnetic ﬁeld (IMF), total IMF (jBj), proton density (Np) and solar wind velocity (Vsw). The IMF parameters used in the study are considered in Geocentric Solar Magnetospheric System (GSM) coordinates in which X-axis is pointed from the Earth to the Sun, the Y-axis is deﬁned to be perpendicular to the Earth’s magnetic dipole so that the X–Z-plane contains the dipole axis and the positive Z-axis is chosen to be in the same sense as the northern magnetic pole. The impact of transient solar emissions on the magnetosphere are evidenced in ground magnetic variations as observed in the magnetic elements namely, horizontal (H), vertical (Z), and declination (D) components of the geomagnetic ﬁeld.

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For the current study the variations in these elements are acquired from the low latitude magnetic observatory, Alibag (geographic latitude 18.63°N, longitude 72.87°E; geomagnetic latitude 10.02°N, longitude 145.97°E). Diurnal characteristics of the two ﬁeld components are represented by DH and DD where; the D symbol marks the departure of the disturbed day variation from quiet day pattern after removing the night base. The night base values for the disturbed day are taken as H and D values on 1900 UT of day before the arrival of interplanetary shock at the magnetopause. Geomagnetic digital data used here is of 1-min resolution. Storm time disturbance index (Dst) is also displayed to highlight the magnitude of storm events, the hourly values are taken from WDC, Kyoto. Energy input from the solar wind into the magnetosphere modiﬁes the state of magnetosphere–ionosphere system at both high and low latitudes. In order to investigate the correspondence between the total energy transferred into the magnetosphere and subsequent variations in high latitude and low latitude electric ﬁelds, magnetic ﬁelds and current systems, the respective quantities have been computed as below. Dawn-dusk interplanetary electric ﬁeld is given as (Burton et al., 1975), Ey ¼ V sw Bz ;

ð1Þ

where Vsw is solar wind velocity and Bz is the north–south component of the interplanetary magnetic ﬁeld. The rate of total solar wind energy input into the magnetosphere as given by empirical relation of Perreault and Akasofu (1978) can be written as, 2

¼ V sw jBj sin4 ðH=2Þ l20 :

ð2Þ

Here, jBj is magnitude of total interplanetary magnetic ﬁeld, H is arctan(By/Bz) for Bz > 0° and 180° – arctan(jByj/jBzj) for Bz 6 0. The ﬁeld-aligned current (FAC) density jk which is controlled by the solar wind is given by the relation proposed by Iijima and Potemera (1982), h i1=2 þ 1:4; ð3Þ jk ¼ 0:0328 N 1=2 p V sw BT sinðH=2Þ where Np is in cm3, Vsw is in km s1; BT ¼ ðB2y þ B2z Þ1=2 is in nT, where Bz and By denote the IMF magnetic components, the angle H is measured between the positive Z-axis and the IMF vector in the Y–Z-plane, and is deﬁned above. Plasma beta is deﬁned as the ratio of plasma thermal pressure and magnetic pressure b¼

N p kBT p ; B2 =2l0

ð4Þ

where kB is the Boltzmann constant = 1.38065 1023 J K1, l0 is permeability of free space, given as 4p 107 N A2. Development of the main phase for intense magnetic storms (Dst 6 100 nT) is guided predominantly by orientation and magnitude of IMF Bz component (Gonzalez

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and Tsurutani, 1987; Gonzalez et al., 1989, 1999, 2002). All the events have been found to exhibit signiﬁcant values of southward oriented Bz after the shock impact. Sharp depression in the horizontal component of geomagnetic ﬁeld, marking the onset of main phase is triggered at the time of southward traversal of Bz. The main phase is deﬁned as the duration between time of sharp depression of Dst index and the time where Dst attains its maximum magnitude (i.e., minimum value on Y-axis in Figs. 1–4) (Gonzalez et al., 1994). 3. Results and discussion The ﬁeld-aligned currents are the main cause of eastward or westward magnetic disturbances in mid-latitudes (Sun et al., 1984), whereas at low and high latitudes the equatorial and auroral electrojets are the prime sources for these variations. From the TRIAD satellite observations two components of the ﬁeld-aligned currents have been discovered, one ﬂows into the ionosphere and the other ﬂows out. By convention, the region on the poleward side is called ‘region 1’ and the equatorward one is ‘region 2’ (Iijima and Potemera, 1976) irrespective of the current ﬂow direction. Region 1 signiﬁes the currents ﬂowing into the ionosphere on the dawnside and out of the ionosphere on the duskside. On the other hand, region 2 ﬁeld-aligned current system lies equatorward of region 1 and has opposite sense of current ﬂow, i.e., with current ﬂowing out of the ionosphere on the dawnside and into the ionosphere on the duskside. Low latitude magnetic ﬁeld variations in response to various interplanetary conditions and their association with ﬁeld-aligned currents during intense (Dstmin 6 200 nT) storm events are discussed here. Reproduced in Figs. 1 and 2 are the solar wind and interplanetary conditions along with the geomagnetic ﬁeld variations as seen at low latitude observatory Alibag during the four intense storm events, respectively. 3.1. 31 March 2001 This is one of the most intense storm events of the solar cycle-23. An intense solar ﬂare (X1.7) occurred at 0957 UT on 29 March 2001, peaking at 1015 UT (from GOES-8 records). Associated with this ﬂare, a halo CME with a speed of 942 km/s was observed by SOHO at 0956 UT on 29 March 2001. After 38 h an interplanetary shock (IPS) was detected by ACE at 0020 UT on 31 March 2001. The shock was marked by abrupt increase in Tp, Np, Vsw and jBj as shown by vertical dashed line in Fig. 1(a). Proton density (Np), solar wind velocity (Vsw) and total IMF (jBj) showed remarkable increase with values 60 cm3, 200 km/s and 30 nT, respectively. Solar wind dynamic pressure increased to values of 36 nPa with further enhancement to values as high as 53 nPa. Increased solar wind dynamic pressure compressed the magnetopause after 32 min of interplanetary shock passage at ACE position, and was observed as sudden

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a

b

30 March-1 April 2001

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Fig. 1. Panels (a) and (b), respectively, are showing the solar wind, interplanetary parameters, Tp, Np, Vsw, Psw, By, Bz, jBj, b, Ey, the rate of energy transferred into the magnetosphere (), the corresponding ground magnetic variation (DHABG) and the Dst for two intense storm events of 31 March 2001 and 20 November 2003. Alibag data have been shifted in accordance to the ACE data. Vertical shaded (light) strips depict the main phase period used and dark shaded portions in By, Ey indicate dusk-ward interval and in Bz show southward interval.

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commencement with amplitude of 140 nT at ground magnetic ﬁeld at 0052 UT on 31 March 2001. Just before

the shock passage IMF By was dusk-ward and Bz was southward for just 1 h, after that By continued dusk-ward

0

nT

-100 -200 -300 -400 12 UT, hrs

though in ﬂuctuating manner. On the other hand, IMF Bz turned northward sharply and continued in that orientation for 2 h (0030–0240 UT). At 0335 UT on 31 March 2001, Bz turned southward abruptly (for 4 h) followed by some spikes (30 nT) in Bz until another striking southward orientation occurred at 1345 UT, after a gap of 6 h. Second Bz southward duration persisted between 1345 and 2140 UT. The two large southward Bz rotations are associated with magnetic cloud structures, MC1 and MC2 which are marked by vertical dashed lines in Fig. 1(a). First magnetic cloud (MC1) was observed between 0505 UT and 1015 UT and second cloud (MC2) was identiﬁed between 1235 and 2140 UT. During MC1, maximum magnetic ﬁeld strength was 49.1 nT, peak southward Bz was 47.9 nT and plasma beta (b) was 0.074 (below

0.1, characteristic of magnetic clouds). Within MC2, maximum jBj, southward Bz and b were 41.4 nT, 36.8 nT and 0.075, respectively. During the two magnetic clouds the proton temperature (Tp) was in the order of 105 K, which is one order higher than the typical magnetic cloud (104 K). The diﬀerence in temperatures is ascertained to be due to compression between the two magnetic clouds (MC1 and MC2) (Wang et al., 2003). Solar wind velocity exhibited multiple hikes throughout the storm main phase period and attained a peak of 900 km/s whereas proton density showed ﬂuctuations dying out with main phase period. A similar two-step change in orientation was observed in dawn-dusk electric ﬁeld. Solar wind energy input ‘’ increased abruptly to 1.7 1013 W with southward shooting of Bz (10 nT) at 0020 UT and subsequently

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increased consistently with orientation of Bz, reaching higher values 3.2 1013 W. The function shows consistency with two step dusk-ward orientation of Ey. Main phase triggering for this storm event was delayed as seen at ground records and can be ascribed to 2 h large (53 nT) northward directed Bz after the shock and this period between 0030 and 0240 UT is speciﬁed as initial phase. Subsequently, a sharp and intense main phase started at 0500 UT on 31 March, under the favourable Bz and solar wind conditions (vertical shaded strip in Fig. 1(a)). In response to second southward orientation of Bz (1345 UT) and substantial energy transfer ( 1 1013 W), the main phase experiences start of another depression (DHABG) at 1700 UT on 31 March 2001 after 8 h of ﬁrst dip. Thus, before complete cessation of main phase a second enhancement in ring current is quite evident for this storm. Dusk-ward turnings of IMF By and southward orientations of IMF Bz are in good correspondence for this storm. This intense magnetic storm is explained as double-peak structured by Xue et al. (2005). However, although the two step features during two periods between 0500 and 0900 UT and 1700–2200 UT on 31 March are observed in various parameters, this storm cannot be qualiﬁed as a two-step main phase storm, as it does not satisfy the criteria proposed by Kamide et al. (1998). The two-step main phase storm proﬁle suggested by Kamide et al. (1998), describes the condition of the minimum Dst as the second minimum being larger in magnitude followed by recovery. The present storm is characterized by the ﬁrst larger minimum Dst, then a partial recovery, followed by the second larger minimum and then the recovery phase. Minimum Dst for this event was 383 nT. 3.2. 20 November 2003 On 18 November 2003, an X-ray solar ﬂare (M3.2) was recorded by GOES-8 at 0723 UT. Following this a halo CME was observed by LASCO/SOHO with a speed of 1660 km/s on 18 November 2003 at 0819 UT. On 20 November 2003, the fast halo CME (Gopalswamy et al., 2005) drove an interplanetary shock at 0720 UT featured by sudden increase in all solar wind parameters and IMF jBj. Fig. 1(b) illustrates proton density (Np), solar wind speed (Vsw) and IMF jBj increased to values 16 cm3, 700 km/s and 22 nT, respectively. The shock was followed by a sheath of shocked interplanetary plasma characterized by increased ﬂuctuating ﬁeld strength (jBj), speed, density and temperature for nearly 3 h 40 min. After 45 min from IPS at ACE, the increase in solar wind dynamic pressure (12 nPa) produced magnetopause compression marked by storm sudden commencement (SSC) at 0805 UT with amplitude 25 nT. The shock driven by ICME contained a magnetic cloud structure (Zhang et al., 2007), characterized by increase in IMF jBj to values as high as 55 nT and large southward Bz prevailing for 13 h with a peak of 50 nT. IMF By attained large val-

ues in dusk-ward direction. Large dusk-ward directed By and southward directed Bz were observed to occur with some time lag. Magnetic cloud (MC) passage was observed from 1100 UT of 20 November to 0000 UT of 21 November, the duration is marked by two big dashed lines in Fig. 1(b). Maximum plasma beta for the cloud duration was 0.075. Peak By (dusk-ward) increased to 40 nT. Epsilon increased to a large value of 2.5 1012 W during the shock passage when Bz was southward (10 nT). Later on with Bz turning largely southward, energy input is enhanced considerably to values as high as 3.5 1013 W. Main phase depression observed in DHABG started shortly after the SSC in consistence with southward Bz and the main phase (1100–1900 UT) prevailed for 8 h followed by rapid recovery. Main phase for this intense storm event developed to a magnitude of 660 nT and Dst attained a peak of 422 nT. Large fraction of solar wind energy input to the magnetosphere under the favourable IMF Bz conditions contributed to the large ring current development for this intense storm event (Dst 422 nT). 3.3. 6 April 2000 A long duration C9.7 solar ﬂare occurred at 1512 UT on 4 April 2000 and peaked at 1541 UT on the same day as recorded by GOES-8. Associated with this ﬂare activity a halo coronal mass ejection was observed by LASCO at 1452 UT on 4 April 2000 with a speed of 1188 km/s. An IPS was recorded by ACE satellite at 1600 UT on 6 April 2000, after 49 h of CME from the Sun. Distinct shock (IPS) is featured by abrupt increase in solar wind parameters and total interplanetary magnetic ﬁeld, jBj (marked by vertical dashed line in Fig. 2(a)). Proton density, solar wind speed and jBj increased to high values of 24 protons cm3 and 560 km/s and 28 nT, respectively. The solar wind dynamic pressure (Psw) rose to values 12 nPa and increased further to higher values as high as 25 nPa. After the shock the temperature and density remained high and the magnetic ﬁeld was ﬂuctuating. As suggested by Tsurutani et al. (1988), such rapidly varying magnetic ﬁeld direction, high temperature and density are characteristic of the sheath region. After about 40 min, the shock impinged the magnetopause and the high solar wind pressure (Psw) lead to magnetopause compression on 6 April 2000 at 1640 UT, as evidenced in the form of sudden jump (storm sudden commencement) in DHABG at low latitude observatory Alibag with amplitude of 50 nT (Fig. 2(a)). The storm on 6– 7 April 2000, was an example of a magnetospheric storm driven by an intense southward IMF in the sheath region between the shock and the ICME (Huttunen et al., 2002; Xue et al., 2005). IMF Bz traversed southward sharply after the shock impact and remained southward for 8 h reaching a peak of 32 nT at 2230 UT on 6 April 2000. This intense southward ﬁeld was located in the shock sheath, which resulted from shock compression. IMF By exhibited ﬂuctuating behaviour for storm duration, though

predominantly dawn-ward. Solar wind speed (Vsw) retained its high level (600 km/s) attained at shock passage time for almost complete duration of storm. Signiﬁcant increase (8 1012 W) in the energy input ‘’ function as computed from Eq. (2) just after the shock passage can be attributed to predominantly southward oriented Bz prior to shock. The increased further to still higher values 1.2 1013 W owing to prolonged southward orientation of Bz. The electric ﬁeld computed from Eq. (1) shows dusk-ward directed ﬁeld for 8 h with average value of 13 mV/m. Southward traversal of IMF Bz triggered the commencement of sharp main phase (vertical shaded strip in Fig. 2(a)) for this storm event. Onset of the storm main phase was at 1700 UT, as seen from Dst index. Main phase period lasted for 7 h between 1700 UT (6 April) and 0000 UT (7 April) attaining a magnitude 330 nT; peak Dst for this event is 288 nT. 3.4. 24 August 2005

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On 22 August 2005, two M-class solar ﬂares with magnitudes M2.6 and M5.6 were recorded at 0044 UT and 1646 UT. Peaks for the two ﬂares were recorded at 0133 UT and 1727 UT, respectively (GOES-8). Successive

to the ﬂares, two full halo CME hurled into the space, moving with high speeds of 1194 km/s and 2378 km/s as observed at 0104 UT and 1705 UT, respectively, on 22 August. At 0535 UT on 24 August 2005, ACE observed a fast forward shock (shock 1), driven by the ﬁrst halo CME of 22 August that occurred at 0104 UT, marked by a sudden increase in Np, Vsw, and jBj to values 23 cm3, 537 km/s and 24 nT, respectively (Fig. 2(b)). Successive to the ﬁrst shock, another shock (shock 2) hit the magnetopause at 0820 UT (as observed at ACE) marked by one more abrupt increase in Np, Vsw, and jBj by 37 cm3, 578 km/s and 42 nT, respectively. Further IMF jBj increased signiﬁcantly to high values of 60 nT. The solar wind signatures are most consistent with the successive arrivals of CMEs associated with the two ﬂares M2 at 0133 UT and M5 at 1727 UT. The period of ﬂuctuating increase in solar wind and interplanetary magnetic ﬁeld parameters mark the sheath of shocked IP plasma. Solar wind dynamic pressure increased to 11 nPa and 21 nPa at the passage of two successive shocks, respectively, and gained higher levels later on upto values 32 nPa. IMF By was dusk-ward prior to shock and persisted dusk-ward after shock passage for 4 h. First interplanetary shock impacted at magnetopause resulting in

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0 20 Ey(mV/m)

40

sudden hike in DHABG known as storm impulse (SI) after about 40 min at 0615 UT on 24 August. During this period IMF Bz was ﬂuctuating subduing main phase development after this shock. Second shock impact at magnetopause lead to SSC of amplitude 30 nT after 30 min of IPS at 0900 UT. During the ﬁrst shock passage, epsilon increased to 1.5 1012 W and for another shock the increase was upto a value of 3.8 1013 W. Larger increase at second shock than during ﬁrst one was aided by large southward Bz (55 nT). Soon after this SSC, depression in DHABG began, marking the main phase onset at 0900 UT. A steep southward traversal in Bz was seen at 0830 UT marked with a peak value of 55 nT followed by a sharp northward turning (1035 UT, peak 23 nT). Sheath of shocked plasma had ICME at the rear end, featured by enhanced magnetic ﬁeld (jBj), smooth rotation in ﬁeld direction both for Bz and By and low plasma beta (b 0.071). Large southward Bz in the ICME assisted in the rapid depression in DHABG and intensiﬁcation of main phase resulting in a magnitude of 310 nT. However, small duration of Bz (2 h) in southward orientation impeded further intensiﬁcation of storm. Peak Dst for this storm event is 216 nT. At 2030 UT on 24 August 2005, a reverse shock is seen marked by increase in Vsw and decrease in jBj and Np. The recovery of this storm event was unsteady and prolonged.

3.5. Variations in DD and ﬁeld-aligned currents Signiﬁcant amplitude variations in east-west component of geomagnetic ﬁeld (DDABG) are obtained for all four storm events consistent with unsteady ﬁeld-aligned current density (jk). Hence, an attempt is made to examine the correspondence between DDABG and jk. Fig. 3 depicts FAC densities (jk), declination (DDABG) and horizontal (DHABG) components of geomagnetic ﬁeld as recorded at Alibag for the two intense storms (described under Sections 3.1 and 3.2). The ﬁeld-aligned current density, jk is computed from Eq. (3). Vertical shaded strip shows the main phase interval for all events. The main phase period for current analysis is considered between the time of sharp depression in Dst index and attainment of highest magnitude of Dst. Chun and Russell (1997) pointed out that during geomagnetic active periods the region of FAC expands and is seen at lower L values particularly in pre-midnight and post-midnight sectors. The storm event of 31 March 2001 was characterized by a sharp increase in jk with values reaching upto 12 lA/m2 at the shock passage time (Fig. 3(a), top panel). Declination component was predominantly eastward during the main phase period (0500–0900 UT and 1700– 2200 UT) and seems to follow two-step feature of Bz (Fig. 3(a), middle panel). Eastward development of decli-

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nation is consistent with main phase development, as seen by DHABG (Fig. 3(a), bottom panel). For the intense storm event of 20 November 2003, jk increased to values 13 lA/m2 further increasing to higher values up to 27 lA/m2 (Fig. 3(b), top panel). DDABG increased to values 40 nT (Fig. 3(b), middle panel). DDABG and jk show good correspondence during main phase interval (1100–1900 UT). Eastward development of declination is distinct with main phase development. Correlation between low latitude declination and high latitude ﬁeld-aligned current density is represented in Fig. 3(c) and (d) for the two intense storm events during the main phase period. For 31 March 2001 event, increase in ﬁeldaligned current density aids the eastward development of declination and a fair correlation is seen (R = 0.58). In case of intense event of 20 November 2003, reverse trend is seen, increasing FAC corresponds to westward declination with a good correlation (R = 0.73). The reverse trend seen for 20 November 2003 is basically associated with the phase shift in the variation trend in jk and DDABG. Variation of FAC, declination and horizontal components for the 6–7 April 2000 and 24–25 August 2005 are shown in Fig. 4. For the storm event of 6 April 2000, jk increased to values 10 lA/m2 at the shock passage time, gained a peak of 11 lA/m2 at 2240 UT and remained at values >5 lA/m2 for a continuous period of 14 h later also (Fig. 4(a), top panel). With the development of the main phase for this storm event, declination component of geomagnetic ﬁeld (DDABG) also increased towards eastward direction (Fig. 4(a), middle panel). The main phase interval for this storm is considered between 1700 UT on 6 April and 0000 UT on 7 April. Storm event of 24 August 2005 exhibits increase in ﬁeldaligned current density after the shock passage to values 8 lA/m2 and 15 lA/m2 at the passage of two successive shocks (Fig. 4(b), top panel). Main phase period (0900– 1100 UT) for this storm was shorter (2 h) as compared to other intense storms in this study, as evident from Dst

a

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Y = 112.9 - 6.9 X + 0.02 X2 R = 0.99

(Fig. 4(b), bottom panel). DDABG shows west-ward development in this interval and was unsteady further (Fig. 4(b), middle panel). Correlations between FAC and DDABG for 6 April 2000 and 24 August 2005 are depicted in Fig. 4(c) and (d). The storm events of 6 April 2000 shows a fair correspondence (R = 0.60) between FAC and declination, consistent with 31 March 2001 storm event. Short duration main phase (2 h) of 24 August 2005 also seem to exhibit good correspondence (R = 0.62) between FAC and DDABG. 3.6. Eﬀect of Ey on DD In order to examine the inﬂuence of interplanetary electric ﬁeld (Ey) on declination component of geomagnetic ﬁeld (DDABG), correlation analysis is carried out for the duration of main phase for the two intense storm events of 31 March 2001 (Fig. 5) and 20 November 2003 (Fig. 6). For both the intense storm events, the declination component seems to follow the electric ﬁeld (Ey) with a time lag (Figs. 5(a) and 6(a)). Lag times computed by cross-correlation analysis for 31 March 2001 and 20 November 2003 events come out to be 80 min and 110 min, respectively. Introduction of this time lag clearly brings out reasonably good linear correspondence between the two parameters with R = 0.66 (Fig. 5(c)) and R = 0.75 (Fig. 6(c)), respectively, for the two events as compared to without lag correlation (Figs. 5(b) and 6(b)). It is quite evident from the respective ﬁgures that complete main phase duration exhibits scatter between the two parameters, whereas in the dusk-ward electric ﬁeld section good correlation is obtained. Our results are in well conformity with Clauer and McPherron (1980) where they have indicated a direct causative role of solar wind electric ﬁeld in the development of low latitude asymmetry. The ﬁeld-aligned current densities attained peak values >10 lA/m2 for all storms, however, the largest ﬁeld-aligned current density (27 lA/m2) is found for intense storm event of 20 November 2003.

b

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3.7. Intensity of storm in response to IMF Bz and energy inﬂux To summarize, the occurrence characteristics of the storms under study, a strong dependence of storm intensity on the duration and the magnitude of the southward directed Bz is seen. Fig. 7(a) depicts the correspondence between the time lag Hlag (expressed as the time when DHABG attained its minimum value during the main phase with respect to the Bzmin ) with the storm intensity Dstmin. It is clear from Fig. 7(a) that larger Hlag leads to more intense magnetic storms, but the relationship is not a linear one. A noticeable feature representing the relation between the magnitudes of energy injected into the magnetosphere with the storm intensity is illustrated in Fig. 7(b). It is noticed that the maximum energy ﬂux () represented by Epsmax shows a well deﬁned linear trend (R = 0.99) with the maximum Dst magnitude (Dstmin) for the three intense storm events of 6 April 2000, 31 March 2001 and 20 November 2003, thus suggesting that the solar wind energy injected into the magnetosphere assists the development of intense magnetic storm. These three storm events are superintense storms (Dst 6 200 nT). Our results are in good conformity with De Lucas et al. (2007), where the total solar wind energy rate () is modiﬁed applying diﬀerent corrections and linear relation is obtained between Dst and energy for superintense storms (Dst 6 200 nT). The storm event of 24 August 2005 (Dst = 216 nT) remains outside of the ﬁt, which may be attributed to the short duration (2 h) of southward orientation of IMF Bz, although its magnitude was as high as 55 nT. It appears that persistence of southward oriented Bz for short period (2 h), could produce a storm of Dst = 216 nT only, which has much less intensity than the other three events, although Bz magnitude was very large. 4. Conclusion A study has been carried out to examine the correspondence between FACs and ground magnetic signatures. For the intense storm event of 20 November 2003, a close correspondence between DDABG and jk clearly advocates the proposition for the low latitude observations for understanding the high latitude and low latitude coupling processes (Chun and Russell, 1997). Correlation between DDABG and Ey for the main phase periods, clearly shows a well deﬁned second-order polynomial ﬁt when electric ﬁeld (Ey) is dusk-ward, whereas a scatter is quite prominent during periods of dawn-ward directed electric ﬁeld. Duskward oriented electric ﬁelds inﬂuence the declination component. When the interplanetary electric ﬁeld is dusk-ward, the declination component of the geomagnetic ﬁeld clearly tends to maintain an eastward response which signiﬁes an equator-ward upper atmospheric current.

Another important result of present study is that, the product of the duration and magnitude of the southward IMF Bz seems to play signiﬁcant role in controlling the intensity of the magnetic storms. Also from the estimate of energy inﬂux into the magnetosphere it is found that, large energy input with suﬃciently long duration southward Bz assists in development of superintense magnetic storms (Dst 6 200 nT). Acknowledgments Authors extend thanks to ACE/MAG/SWEPAM instrument teams for providing the interplanetary magnetic ﬁeld and solar wind data. Thanks are due to GOES-8 instrument team members for providing solar ﬂare data. We acknowledge WDC, Kyoto for providing Dst indices. We also thank NASA website from where coronal mass ejection information was extracted. This CME catalog is generated and maintained by NASA and The Catholic University of America in cooperation with the Naval Research Laboratory. G.S.L. would like to thank the Indian National Science Academy, New Delhi for the support under the Senior Scientist scheme. References Akasofu, S.-I. Energy coupling between the solar wind and the magnetosphere. Space Sci. Rev. 28, 121–190, 1981. Akasofu, S.-I., Chapman, S. On the asymmetric development of magnetic storm ﬁeld in low and middle latitudes. Planet. Space Sci. 12, 607–626, 1964. Baker, D., Zwickl, R., Bame, S., Hones Jr., E., Tsurutani, B.T., Smith, E., Akasofu, S.-I. An ISEE-3 high time resolution study of interplanetary parameter corrections with magnetospheric activity. J. Geophys. Res. 88, 6230–6242, 1983. Birkeland, K. The Norwegian Aurora Polaris Expedition 1902-3, On the cause of Magentic Storms and the Origin of Terrestrial Magnetism, vol. 1, H. Aschehoug, Christiania, Norway, 1908. Burton, R.K., McPherron, R.L., Russell, C.T. An empirical relationship between interplanetary conditions and Dst. J. Geophys. Res. 80, 4204– 4214, 1975. Cahill Jr., L.J. Inﬂation of the inner magnetosphere during a magnetic storm. J. Geophys. Res. 71, 4505–4519, 1966. Chen, M.W., Schultz, M., Lyons, L.R., Gorney, D.J. Stormtime transport of ring current and radiation belt ions. J. Geophys. Res. 98, 3835–3849, 1993. Chun, F.K., Russell, C.T. Field-currents in the magnetosphere in the inner magnetosphere: control by geomagnetic activity. J. Geophys. Res. 102, 2261–2270, 1997. Clauer, C.R., McPherron, R.L. The relative importance of the interplanetary electric ﬁeld and magnetospheric substorms on partial ring current development. J. Geophys. Res. 85, 6747–6759, 1980. Crooker, N.U., Siscoe, G.L. Birkeland currents as the cause of the lowlatitude asymmetric disturbance ﬁeld. J. Geophys. Res. 86, 11201– 11210, 1981. Daglis, I.A., Kozyra, J.U. Outstanding issues of ring current dynamics. J. Atmos. Terr. Phys. 64, 253–264, 2002. Daglis, I.A., Thorne, R.M., Baumjohann, W., Orsini, S. The terrestrial ring current: origin, formation, and decay. Rev. Geophys. 37, 407–438, 1999. De Lucas, A., Gonzalez, W.D., Echer, E., Guarnieri, F.L., Dal Lago, A., Da Silva, M.R., Vieira, L.E.A., Schuch, N.J. Energy balance during

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Advances in Space Research 43 (2009) I www.elsevier.com/locate/asr

List of Referees We thank the following individuals for refereeing the papers in this issue of ASR. In addition to the names listed, there were a number of individuals who wished to remain anonymous. Luciano Anselmo

Dimitris Kaskaoutis

Nanan Balan

Karl-Ludwig Klein

Timothy S. Bastian

S. M. Krimigis

Slawomir Breiter

Wei-Tou Ni

R. A. Burger

R. Pinker

G. P. Chernov

J. Schott

Richard Crowther

A. N. Swamy

Jose A. de Diego

Pascal Willis

Hansjoerg Dittus

D. Winske

E. Echer

I. Wytrzyszczak

doi:10.1016/S0273-1177(09)00260-9

Author Index Alex, S., 1575 Alexeev, I.V., 1588 Anselmo, L., 1491

Lakhina, G.S., 1575 Langmayr, D., 1588 Lemaıˆtre, A., 1509 Levy, A., 1538 Lui, A.T.Y., 1588

Badarinath, K.V.S., 1545 Be´rio, P., 1538 Bhattacharya, Amit K., 1563 Biernat, H.K., 1588

Majumdar, T.J., 1563 Mester, John, 1532 Me´sza´rosova´, H., 1479 Me´tris, G., 1538 Mu¨hlbachler, S., 1588

Carletti, T., 1509 Cecatto, J.R., 1479 Christophe, B., 1538 Courty, J.-M., 1538

Overduin, James, 1532

Daly, P.W., 1588 de Andrade, M.C., 1479 Delsate, N., 1509

Pardini, C., 1491 Ragot, B.R., 1484 Rawat, R., 1575 Reynaud, S., 1538 Ryba´k, J., 1479

Erkaev, N.V., 1588 Everitt, Francis, 1532

Gloeckler, G., 1471

Sawant, H.S., 1479 Sharma, Anu Rani, 1545 Singh, A.K., 1555 Srivastava, P.K., 1563

Huai-rong, Shen, 1527

Valk, S., 1509

Jirˇicˇka, K., 1479

Worden, Paul, 1532

Kahler, S.W., 1484 Karlicky´, M., 1479 Kharol, Shailesh Kumar, 1545 Kumar, Sanjay, 1555

Yi-yong, Li, 1527

Fernandes, F.C.R., 1479 Fisk, L.A., 1471

Zhi, Li, 1527

III

Available online at www.sciencedirect.com

Advances in Space Research 43 (2009) 1588–1593 www.elsevier.com/locate/asr

Cluster observations showing the indication of the formation of a modiﬁed-two-stream instability in the geomagnetic tail S. Mu¨hlbachler a,*, D. Langmayr b,1, A.T.Y. Lui c, N.V. Erkaev d, I.V. Alexeev e, P.W. Daly a, H.K. Biernat b a

Max Planck Institut fu¨r Sonnensystemforschung, Max-Planck-Strasse 2, 37191 Katlenburg/Lindau, Germany b Space Research Institute, Austrian Academy of Sciences, Schmiedlstrasse 6, 8042 Graz, Austria c John Hopkins University, Applied Physics Laboratory, 11100 John Hopkins Road, 20723 Laurel, MD, USA d Institute of Computational Modelling, Russian Academy of Sciences, 660036 Krasnoyarsk 36, Russia e Mullard Space Science Laboratory, University College, RH5 6NT Surrey, London, UK Received 15 November 2006; received in revised form 12 December 2008; accepted 14 January 2009

Abstract This study presents several observations of the Cluster spacecraft on September 24, 2003 around 15:10 UT, which show necessary prerequisites and consequences for the formation of the so-called modiﬁed-two-stream instability (MTSI). Theoretical studies suggest that the plasma is MTSI unstable if (1) a relative drift of electrons and ions is present, which exceeds the Alfve`n speed, and (2) this relative drift or current is in the cross-ﬁeld direction. As consequences of the formation of a MTSI one expects to observe (1) a ﬁeld-aligned electron beam, (2) heating of the plasma, and (3) an enhancement in the B-wave spectrum at frequencies in the range of the lower-hybridfrequency (LHF). In this study we use prime parameter data of the CIS and PEACE instruments onboard the Cluster spacecraft to verify the drift velocities of ions and electrons, FGM data to calculate the expected LHF and Alfve`n velocity, and the direction of the current. The B-wave spectrum is recorded by the STAFF instrument of Cluster. Finally, a ﬁeld aligned beam of electrons is observed by 3D measurements of the IES instrument of the RAPID unit. Observations are veriﬁed using a theoretical model showing the build-up of a MTSI under the given circumstances. Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Cluster; Energetic particles; Magnetotail; Modiﬁed-two-stream instability

1. Introduction and methodology There are many occasions in space plasmas where electrons and ions have a relative velocity towards each other. Such a current is the source of several kinetic instabilities arising in the plasma. Among them, the modiﬁed-twostream instability (MTSI) plays a crucial role providing a possible explanation for some energetic phenomena as observed in the neighborhood of the Earth. The MTSI occurs in several regions in the terrestrial neighborhood. *

Corresponding author. E-mail address: [email protected] (S. Mu¨hlbachler). 1 Present address: The Virtual Vehicle Competence Center (vif), Inﬀeldgasse 21a, 8010 Graz, Austria.

For instance, Winske et al. (1985, 1987) successfully identiﬁed the MTSI by comparing the macroscopic consequences of this instability with data obtained from ISEE at the Earth’s bow shock. Performing full particle simulations, Matsukiyo and Scholer (2003) and Scholer et al. (2003) also revealed the excitation of the MTSI at the bow shock. Another important application ﬁeld of the MTSI is the terrestrial magnetotail. In particular in the context of magnetospheric substorms the role of the so-called cross-ﬁeld current instabilities (CCI), which is an umbrella term for several branches of solutions of a dispersion relation including the MTSI, the lower hybrid drift instability (LHDI), and the ion-Weibel instability (IWI), was discussed by many authors (see e.g., Yoon et al., 2002; Lui, 2004, and references cited therein). In the magnetotail,

0273-1177/$36.00 Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2009.01.012

S. Mu¨hlbachler et al. / Advances in Space Research 43 (2009) 1588–1593

the current sheet separating the oppositely oriented magnetic ﬁelds is the source for the CCI. Both, the MTSI and the LHDI, are operative in the same frequency range. However, as demonstrated by Yoon and Lui (2004), the MTSI is probably the most dominant instability in the center of the current sheet where the current is very strong, whereas at the ﬂanks of the sheet, the LHDI driven by density gradients overtakes the MTSI (Yoon et al., 2002). Thus, on basis of the results of the above studies, we summarize the characteristics of the MTSI. First of all we shall observe a relative velocity diﬀerence between electrons and ions leading to a cross-ﬁeld current. In addition the relative velocity between those two populations has to exceed the local Alfve`n velocity. The lower the observed plasma beta the stronger is the evolution of the MTSI. Once the MTSI has established it produces waves in the range of the lowerhybrid frequency, a heating of the plasma takes place and an acceleration of electrons in a ﬁeld aligned direction. In the following sections of this paper, we study Cluster observations of September 24, 2003, between 15:00 UT and 15:25 UT, which show either the requisites for or the eﬀects of the formation of the MTSI. Finally, we discuss possible explanations of this fact for dynamics in the geomagnetic tail. 2. Cluster position and conﬁguration During the event we are studying the Cluster ﬂeet is on a tailward pass close to the neutral sheet on the duskward ﬂank of the magnetosphere (see Fig. 1). On the top right corner of each picture (left: meridional plane; right: equatorial plane) the conﬁguration of the four spacecraft is drawn. The green and blue2 lines display the actual bow shock and magnetopause positions calculated with existing models of Farris and Russell (1994) and Shue et al. (1998), respectively, and using hourly averages of the z-component of the interplanetary magnetic ﬁeld (IMF Bz) and the solar wind dynamic pressure (Pdyn) as recorded by ACE SWE and MFI instruments, with Bz = 1.8 nT and Pdyn = 4.05 nPa. The orange line shows the orbit of Cluster reference spacecraft (C3) with its position at 15:10 UT marked by the green triangle. The four spacecraft are separated between 50 and 250 km. 3. Cluster magnetic ﬁeld, plasma and wave observations Fig. 2 shows from top to the bottom the magnetic ﬁeld vector (Bx, By, Bz) and the total magnetic ﬁeld in GSE coordinates as observed by the FGM instruments of Clusters 1, 3, and 4. Whereas the black line represents data of C1, green C3 and magenta C4.3 We do not show observations 2 For interpretation of the references to color in Fig. 1, the reader is referred to the web version of this paper. 3 For interpretation of the references to color in Fig. 2, the reader is referred to the web version of this paper.

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of Cluster 2 as there are no CIS measurements available on Cluster 2. Parallel to the magnetic ﬁeld situation we analyze the plasma properties of ions and electrons as observed by the CIS and PEACE instruments of the Cluster spacecraft. Table 1 lists the magnetic ﬁeld and plasma parameters during the period of interest 15:10:56–15:11:45 UT (column 1). The adjacent columns give from left to right the total magnetic ﬁeld, the x-, y-, and z-component of the magnetic vector (all in nT), the electron (Ne) and ion (Ni) density in m 3. The last but one column shows the total relative velocity of ions and electrons, calculated as Vdiﬀ = ((Vxi Vxe)2 + (Vyi Vye)2 + (Vzi Vze)2)1/2. Finally, in the last column we see the angle between this diﬀerential velocity and the ambient magnetic ﬁeld. Taking a closer look onto these parameters we identify a density minimum around the center of our interval. Judging from the above ﬁeld and plasma parameters we suppose that the neutral sheet was crossed between 15:11 and 15:12 UT (grey shaded) when a dipolarization of the current sheet already started. The velocity diﬀerence exceed well the local Alfve`n velocity (as reference we take the local proton Alfve`n velocity, vA), which is the most important requisite for the plasma to become MTSI unstable. And ﬁnally, the calculated angle reﬂects a current in a nearly cross-ﬁeld direction. Assumed that a MTSI builds up, we expect to observe waves in the range of the low hybrid frequency, which we calculated for our circumstances to 10 Hz. Fig. 3 shows the log power spectral densities for C1 in the upper panel and the log power for 1–10 Hz for C1–C4 in the bottom one. We can see a clear enhancement on the log power density spectra between the red vertical lines in the frequency range we expect. (The ﬁgure is on courtesy of STAFF-PI.)

4. RAPID electron observations During the event we are studying the electron instrument of the RAPID unit is in burst mode, which oﬀers the possibility to display a full 3D picture of electron ﬂuxes with a 4s time resolution. In Fig. 4 we present the electron pitch angle distribution of C1 in the lowest energy channel, which covers a range of 35–51 keV. Electron pitch angles are arranged in a 5° step. At around 15:16 UT we recognize a change in the pitch angle distribution from a quasi isotropic state to a ﬁeld aligned distribution. We augment this picture by studying the 3D ﬂow direction as presented in Fig. 5. This ﬁgure shows a sequence of 16 3D ﬂow pictures in a so-called bispherical view, whereas the left sphere reﬂects a view at top of the spacecraft and the right sphere at the bottom. Black dots mark the ﬂow direction 90° to the magnetic ﬁeld and the red4 star indicates the positive magnetic ﬁeld 4 For interpretation of the references to color in Fig. 5, the reader is referred to the web version of this paper.

vector with the red dot showing the opposite. The magnetic ﬁeld vector has a nearly dipolar orientation. The series of ﬂow pictures shows a peak of ﬂux of electrons just along the ﬁeld vector starting at 15:16 UT.

5. Summary and conclusion We study data of September 24, 2003, between 15:00 and 15:25 UT with respect to theoretical studies, which

S. Mu¨hlbachler et al. / Advances in Space Research 43 (2009) 1588–1593

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Table 1 Magnetic ﬁeld and plasma parameters during the interval of interest 15:10:56–15:11:45 UT. Time (UT)

Btot (nT)

15:10:56.979 15:11:01.014 15:11:05.049 15:11:17.152 15:11:21.187 15:11:29.256 15:11:33.291 15:11:37.325 15:11:41.360 15:11:45.395

17.44 7.71 12.90 14.73 12.43 16.24 16.50 13.87 13.72 12.38

Bx (nT)

By (nT)

3.89 3.25 7.62 3.01 4.29 2.56 5.65 2.82 4.4 3.87

5.51 1.54 3.97 1.75 0.15 1.53 3.24 5.74 8.14 6.93

Bz (nT)

Ne (m 3)

Ni (m 3)

Vdiﬀ (km s 1)

h (°)

16.08 6.82 9.63 14.31 11.66 15.96 15.16 12.31 10.13 9.50

0.18 0.24 0.17 0.13 0.15 0.14 0.15 0.14 0.13 0.22

0.14 0.13 0.11 0.1 0.08 0.07 0.12 0.10 0.14 0.16

5518.78 1906.14 1586.11 2239.61 1986.86 2284.30 742.43 1118.11 2110.63 1697.87

76.98 111.12 129.73 105.95 114.77 88.06 107.71 97.57 71.48 44.66

2

Log Power Spectral density, (nT) /Hz Data from step5: Calibrated data in SR2 system [nT] with DC (0.00-225.00Hz) N = 8192 dt=18.204 sdf=0.0549 HzFc=0.10 F1=0.00 F2=225.00

Frequency (Hz)

Rumba (1)

200 150 100 50 0 14:50

15

1.0 - 10.0 Hz

Log power, nT2

14:50

15:20

15:10

15:20

-2 -3 -4 -5 14:50

U.T.

15:10

15

0 -1

14:50

15

15:10

15

15:10

15:20

15:20

24/September/2003 Fig. 3. B-Wave spectrogram of STAFF (on courtesy of the STAFF-PI).

Fig. 4. Electron pitch angle distribution of C1 in the lowest energy channel.

describe the build up of the so-called modiﬁed-two-streaminstability. Analysis of magnetic ﬁeld data, electron and ion motions, and wave activity conﬁrm theoretically developed requisites for the formation of the MTSI. About 5 min later, which is about 2 min more then the calculated growth time of the instability, the high energy particle experiment RAPID observes a ﬁeld aligned beam, as a consequence of the MTSI build up. Theoretical analysis with respect to the

observed data conﬁrms our observations. During the last years the Cluster mission lead to a lot of important contributions in the investigation of magnetospheric dynamics. However, there are still a bunch of open questions seeking for answers. One of these deals with the fact, where do such ﬁeld aligned high energetic electron beams as reported in this study come from and how are the energized. We believe that the presented work can be a possible scenario

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for this phenomenon. The shown features show a high possibility that the so-called modiﬁed-two-stream instability plays an important role in near tail processes. Ji et al. (2004) study electromagnetic ﬂuctuations during fast reconnection in laboratory plasmas and ﬁnd a positive correlation between these ﬂuctuations up to the range of the LHF (as driven by a MTSI) and fast reconnection. However, to ﬁnish the picture and to strengthen our conclusions a few more events like this have to be studied using more detailed analysis tools such as the curlometer method, which allows the estimation of the current normal to the magnetic ﬁeld (see e.g., Dunlop and Woodward, 2000).

Acknowledgments For the provision of prime parameter data the authors thank the instrument teams of Cluster-CIS, -FGM, -PEACE, and -STAFF, in particular I. Danduras, E. Lucek, A. Fazakerley, and N. Cornilleau-Wehrlin. This work is partly supported by ESTEC Contract 18.201/04/ NL/NR, by DLR Grant 50 OC 0003, by RFBR Grant

Sun View from South

04-05-64088, by Programs 2.17 and 16.3 of RAS, and by Project P17100-N08 of the Austrian Science Fund. References Dunlop, M.W., T. I. Woodward, Cluster magnetic ﬁeld analysis techniques. In: Robert A. Harris (Ed.), Proceedings of the Workshop ‘‘Cluster-II Workshop: Multiscale/Multipoint Plasma Measurements” held at Imperial College, London, UK, 22–24 September 1999, Paris, ESA-SP, vol. 449, pp. 351–354, ISBN: 9290927968, 2000. Farris, M.H., Russell, C.T. Determining the standoﬀ distance of the bow shock: Mach number dependence and use of models. J. Geophys. Res. 99, 17681–17689, 1994. Ji, H., Terry, S., Yamada, M., Kulsrud, R., Kuritsyn, A., Ren, Y. Electromagnetic ﬂuctuations during fast reconnection in a laboratory plasma, Phys. Rev. Lett., 92, 115001, doi:10.1103/PhysRevLett.92.115001, 2004. Lui, A.T.Y. Potential plasma instabilities for substorm expansion onset. Space Sci. Rev. 113, 127–206, 2004. Matsukiyo, S., Scholer, M. Modiﬁed two-stream instability in the foot of a high Mach number quasi-perpendicular shocks, J. Geophys. Res. 108, 1459–1468, doi:10.1029/2003JA010080, 2003. Scholer, M., Shinohara, I., Matsukiyo, S. Quasi-perpendicular shocks: length scale of the cross-shock potential, shock reformation, and

S. Mu¨hlbachler et al. / Advances in Space Research 43 (2009) 1588–1593 implication for shock surﬁng, J. Geophys. Res. 108, 1014–1018, doi:10.1029/2002JA009515, 2003. Shue, J.-H., Song, P., Russell, C.T., Steinberg, J.T., Chao, J.K., Zastenker, G., Vaisberg, O.L., Kokubun, S., Singer, H.J., Detman, T.R., Kawano, H. Magnetopause location under extreme solar wind conditions. J. Geophys. Res. 103, 17691– 17700, 1998. Winske, D., Tanaka, M., Wu, C.S., Quest, K.B. Plasma heating at collision shocks due to the kinetic cross-ﬁeld streaming instability. J. Geophys. Res. 90, 123–136, 1985.

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Winske, D., Giacalione, J., Thomsen, M.F., Mellott, M.M. A comparative study of plasma heating by ion acoustic and modiﬁed two-stream instabilities at subcritical quasi-perpendicular shocks. J. Geophys. Res. 92, 4411–4422, 1987. Yoon, P.H., Lui, A.T.Y. Lower hybrid drift and modiﬁed two stream instabilities in current sheet equilibrium. J. Geophys. Res. 109, doi:10.1029/2003JA010180, 2004. Yoon, P.H., Lui, A.T.Y., Sitnov, M.I. Generalized lower-hybrid drift instabilities in current-sheet equilibrium. Phys. Plasmas 9, 1526–1538, 2002.